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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 348701 6 pageshttpdxdoiorg1011552013348701
Research ArticleSome Remarks on the Extended Hartley-Hilbert andFourier-Hilbert Transforms of Boehmians
S K Q Al-Omari1 and A KJlJccedilman2
1 Department of Applied Sciences Faculty of Engineering Technology Al-Balqarsquo Applied University Amman 11134 Jordan2Department of Mathematics and Institute of Mathematical Research Universiti Putra Malaysia (UPM)43400 Serdang Selangor Malaysia
Correspondence should be addressed to A Kılıcman akilicmanputraupmedumy
Received 19 January 2013 Accepted 15 March 2013
Academic Editor Mustafa Bayram
Copyright copy 2013 S K Q Al-Omari and A Kılıcman This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
We obtain generalizations of Hartley-Hilbert and Fourier-Hilbert transforms on classes of distributions having compact supportFurthermore we also study extension to certain space of Lebesgue integrable Boehmians New characterizing theorems are alsoestablished in an adequate performance
1 Introduction
The classical theory of integral transforms and their applica-tions have been studied for a long time and they are appliedin many fields of mathematics Later after [1] the extensionof classical integral transformations to generalized functionshas comprised an active area of research Several integraltransforms are extended to various spaces of generalizedfunctions distributions [2] ultradistributions Boehmians[3 4] and many more
In recent years many papers are devoted to those integraltransforms which permit a factorization identity (of Fourierconvolution type) such as Fourier transform Mellin trans-form Laplace transform and few others that have a lot ofattraction the reason that the theory of integral transformsgenerally speaking became an object of study of integraltransforms of Boehmian spaces
The Hartley transform is an integral transformation thatmaps a real-valued temporal or spacial function into a real-valued frequency function via the kernel
119896 (120592 119909) = cas (120592119909) (1)
This novel symmetrical formulation of the traditional Fouriertransform attributed to Hartley 1942 leads to a parallelismthat exists between a function of the original variable and thatof its transform In any case signal and systems analysis and
design in the frequency domain using the Hartley transformmay be deserving an increased awareness due to the necessityof the existence of a fast algorithm that can substantiallylessen the computational burden when compared to theclassical complex-valued fast Fourier transform
The Hartley transform of a function 119891(119909) can beexpressed as either [5]
A (120592) =1
radic2120587int
infin
minusinfin
119891 (119909) cas (120592119909) 119889119909 (2)
or
A (119891) = intinfin
minusinfin
119891 (119909) cas (2120587119891119909) 119889119909 (3)
where the angular or radian frequency variable 120592 is related tothe frequency variable 119891 by 120592 = 2120587119891 and
A (119891) = radic2120587A (2120587119891) = radic2120587A (120592) (4)
The integral kernel known as cosine-sine function isdefined as
cas (120592119909) = cos 120592119909 + sin (120592119909) (5)
Inverse Hartley transform may be defined as either
119891 (119909) =1
radic2120587int
infin
minusinfin
A (120592) cas (120592119909) 119889120592 (6)
2 Abstract and Applied Analysis
or
119891 (119909) = int
infin
minusinfin
A (119891) cas (2120587119891119909) 119889119891 (7)
The theory of convolutions of integral transforms has beendeveloped for a long time and is applied in many fields ofmathematics Historically the convolution product [2]
(119891 lowast 119892) (119910) = int
infin
minusinfin
119891 (119909) 119892 (119909 minus 119910) 119889119910 (8)
has a relationship with the Fourier transform with the factor-ization property
F (119891 lowast 119892) (119910) = F (119891) (119910)F (119892) (119910) (9)
Themore complicated convolution theorem of Hartley trans-forms compared to that of Fourier transforms is that
A (119891 lowast 119892) (119910) =1
2G (A119891 timesA119892) (119910) (10)
where
G (119891 lowast 119892) (119910) = 119891 (119910) 119892 (119910) + 119891 (119910) 119892 (minus119910)
+ 119891 (119910) 119892 (119910) minus 119891 (minus119910) 119892 (minus119910)
(11)
Some properties of Hartley transforms can be listed asfollows
(i) Linearity if 119891 and 119892 are real functions then
A (119886119891 + 119887119892) (119910) = 119886A (119891) (119910) + 119887A (119892) (119910) 119886 119887 isinR
(12)
(ii) Scaling if 119891 is a real function then
int
infin
minusinfin
119891 (120572120589) cas (2120587119910120589) 119889120589 = 1119886(A119891) (
119910
119886) (13)
2 Distributional Hartley-Hilbert Transformof Compact Support
TheHilbert transform via the Hartley transform is defined by[6 7]
BA(119910) = minus
1
120587int
infin
0
(A119900
(119909) cos (119909119910) +A119890 (119909) sin (119909119910)) 119889119909
(14)
where
A119900
(119909) =A (119909) minusA (minus119909)
2
A119890
(119909) =A (119909) +A (minus119909)
2
(15)
are the respective odd and even components of (2)We denote C(R) C(R) = C the space of smooth
functions and C1015840(R) C1015840(R) = C1015840 the strong dual of C ofdistributions of compact support overR
Following is the convolution theorem ofBA
Theorem 1 (ConvolutionTheorem) Let 119891 and 119892 isin C then
BA(119891 lowast 119892) (119910) = int
infin
0
(1198961(119909) cos (119910119909) + 119896
2(119909) sin (119910119909)) 119889119909
(16)
where
1198961(119909) = A
119890
119891 (119909)A119900
119892 (119909) +A119900
119891 (119909)A119890
119892 (119909)
1198962(119909) = A
119890
119891 (119909)A119890
119892 (119909) minusA119900
119891 (119909)A119900
119892 (119909)
(17)
Proof To prove this theorem it is sufficient to establish that
1198961(119909) = A
119900
(119891 lowast 119892) (119909) (18)
1198962(119909) = A
119890
(119891 lowast 119892) (119909) (19)
Therefore we have
A119900
(119891 lowast 119892) (119909)
= int
infin
minusinfin
(int
infin
minusinfin
119891 (120574) 119892 (119910 minus 120574) 119889119903) cas (119909119910) 119889119910
= int
infin
minusinfin
119891 (120574)int
infin
minusinfin
119892 (119910 minus 120574) cas (119909119910) 119889119910119889119903
(20)
The substitution 119910 minus 120574 = 119911 and using of (1) together withFubini theorem imply
A119900
(119891 lowast 119892) (119909)
= int
infin
minusinfin
119891 (120574)int
infin
minusinfin
119892 (119911) (cos (119909 (119911 + 120574)) + sin (119909 (119911 + 120574)))
times 119889119911119889119903
(21)
By invoking the formulae
cos (119909 (119911 + 120574)) = cos (119909119911) cos (119909120574) minus sin (119909119911) sin (119909120574)
sin (119909 (119911 + 120574)) = sin (119909119911) cos (119909120574) + cos (119909119911) sin (119909120574) (22)
then (18) follows from simple computation Proof of (19) hasa similar techniqueHence the theorem is completely proved
It is of interest to know that cos(119909119910) and sin(119909119910) aremembers of C and therefore A119900119891A119890119891 isin C1015840 This leads tothe following statement
Definition 2 Let 119891 isin C1015840 then we define the distributionalHartley-Hilbert transform of 119891 as
B
A119891 (119910) = ⟨A
119900
119891 (119909) cos (119909119910)⟩ + ⟨A119890119891 (119909) sin (119909119910)⟩ (23)
The extended transformBA119891 is clearly well defined for
each 119891 isin C1015840
Abstract and Applied Analysis 3
Theorem 3 The distributional Hartley-Hilbert transformBA119891 is linear
Proof Let 119891 119892 isin C1015840 then their components A119890119891A119900119891A119890119892A119900119892 isin C1015840 HenceB
A(119891 + 119892) (119910) = ⟨A
119900
(119891 + 119892) (119909) cos (119909119910)⟩
+ ⟨A119890
(119891 + 119892) (119909) sin (119909119910)⟩ (24)
By factoring and rearranging components we get thatB
A(119891 + 119892) (119910) =
B
A119891 (119910) +
B
A119892 (119910) (25)
FurthermoreB
A(119896119891) (119910) = ⟨119896A
119900
119891 (119909) cos (119909119910)⟩
+ ⟨119896A119890
119891 (119909) sin (119909119910)⟩ (26)
HenceB
A(119896119891) (119910) = 119896
B
A119891 (119910) (27)
This completes the proof of the theorem
Theorem 4 Let 119891 isin C1015840 thenBA119891 is a continuous mapping
onC1015840
Proof Let119891119899 119891 isin C1015840 119899 isinN and119891
119899rarr 119891 as 119899 rarr infinThen
B
A119891119899(119910) = ⟨A
119900
119891119899(119909) cos (119909119910)⟩ + ⟨A119890119891
119899(119909) sin (119909119910)⟩
997888rarr ⟨A119900
119891 (119909) cos (119909119910)⟩ + ⟨A119890119891 (119909) sin (119909119910)⟩
=B
A119891 (119910) as 119899 997888rarr infin
(28)
Hence we have the following theorem
Theorem 5 The mappingBA119891 is one-to-one
Proof Let 119891 119892 isin C1015840 and thatBA119891 =BA119892 then using (23)
we get
⟨A119900
119891 (119909) cos119909119910⟩ + ⟨A119890119891 (119909) sin119909119910⟩
= ⟨A119900
119892 (119909) cos119909119910⟩ + ⟨A119890119892 (119909) sin119909119910⟩ (29)
Basic properties of inner product implies
⟨A119900
119891 (119909) minusA119900
119892 (119909) cos (119909119910)⟩
+ ⟨A119890
119891 (119909) minusA119890
119892 (119909) sin (119909119910)⟩ = 0(30)
Hence
A119900
119891 (119909) = A119900
119892 (119909) A119890
119891 (119909) = A119890
119892 (119909) (31)
ThereforeA119891 (119909) = A
119900
119891 (119909) +A119890
119891 (119909)
= A119900
119892 (119909) +A119890
119892 (119909) = A119892 (119909)(32)
for all 119909 This completes the proof of the theorem
Theorem 6 Let 119891 isin C1015840 then 119891 is analytic and
D119896
119910
B
A119891 (119910) = ⟨A
119900
119891 (119909) D119896
119910cos (119909119910)⟩
+ ⟨A119890
119891 (119909) D119896
119910sin (119909119910)⟩
(33)
Proof of this theorem is analogous to that of the previoustheorem and is thus avoided
Denote by 120575 the dirac delta function then it is easy to seethat
A119890
120575 (119910) = 1 A119890
120575 (119910) = 0 (34)
3 Lebesgue Space of Boehmians forHartley-Hilbert Transforms
The original construction of Boehmians produce a concretespace of generalized functions Since the space of Boehmianswas introduced many spaces of Boehmians were defined Inreferences we list selected papers introducing different spacesof Boehmians One of the main motivations for introducingdifferent spaces of Boehmians was the generalization ofintegral transforms The idea requires a proper choice of aspace of functions for which a given integral transform iswell defined a choice of a class of delta sequences that istransformed by that integral transform to a well-behavedclass of approximate identities and finally a convolutionproduct that behaves well under the transform If theseconditions are met the transform has usually an extensionto the constructed space of Boehmians and the extension hasdesirable properties For general construction of Boehmianssee [8ndash12]
LetD be the space of test functions of bounded supportBy delta sequence we mean a subset of D of sequences (120575
119899)
such that
int
infin
minusinfin
120575119899(119909) 119889119909 = 1
10038171003817100381710038171205751198991003817100381710038171003817 = int
infin
minusinfin
1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 ltM 0 ltM isinR
lim119899rarrinfin
int|119909|gt120576
1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 = 0 for each 120576 gt 0
(35)
where 120576(120575119899)(119909) = 119909 isinR 120575
119899(119909) = 0
The set of all such delta sequences is usually denoted asΔEach element in Δ corresponds to the dirac delta function 120575for large values of 119899
Proposition 7 Let (120575119899) isin Δ then
A119890
120575119899(119910) = int
infin
minusinfin
120575119899(119909) cos (119909119910) 119889119909 997888rarr 1 as 119899 997888rarr infin
A119900
120575119899(119910) = int
infin
minusinfin
120575119899(119909) sin (119909119910) 119889119909 997888rarr 0 as 119899 997888rarr infin
(36)
Let L1(R) L1(R) = L1 be the space of complexvalued Lebesgue integrable functions From Proposition 7 weestablish the following theorem
4 Abstract and Applied Analysis
Theorem 8 Let 119891 isin L1 then BA(119891 lowast 120575
119899)(119910) rarr BA
119891(119910)
as 119899 rarr infin
Proof For 119891 isinL1 (120575119899) isin Δ then using of (14) implies
BA(119891 lowast 120575
119899) (119910)
= int
infin
minusinfin
(A119900
(119891 lowast 120575119899) (119909) cos119909119910 +A119890 (119891 lowast 120575
119899) (119909) sin119909119910) 119889119909
(37)
Since
(119891 lowast 120575119899) (120577) = int
infin
minusinfin
119891 (119905) 120575119899(120577 minus 119905) 119889119905 997888rarr 119891 (120577) (38)
as 119899 rarr infin we see that
A119900
(119891 lowast 120575119899) (119909) = int
infin
minusinfin
(119891 lowast 120575119899) (120577) sin119909120577119889120577
= int
infin
minusinfin
119891 (119905) int
infin
minusinfin
120575119899(120577 minus 119905) sin (119909120577) 119889120577119889119905
997888rarr int
infin
minusinfin
119891 (119905) sin (119909119905) 119889119905
= A119900
(119891) (119909)
(39)
Similarly
A119890
(119891 lowast 120575119899) (119909) 997888rarr A
119890
(119891) (119909) as 119899 997888rarr infin (40)
Therefore invoking the above equations in (37) we getBA(119891 lowast 120575
119899)(119910) rarr BA
119891(119910) as 119899 rarr infin Hence thetheorem
Denote by 120588L1 the space of integrable Boehmians then120588L1 is a convolution algebra when multiplication by scalaraddition and convolution is defined as [9]
119896 [119891119899
120575119899
] = [119896119891119899
120575119899
]
[119891119899
120575119899
] + [119892119899
120574119899
] = [119891119899lowast 120574119899+ 119892119899lowast 120575119899
120575119899lowast 120574119899
]
[119891119899
120575119899
] lowast [119892119899
120574119899
] = [119891119899lowast 119892119899
120575119899lowast 120574119899
]
(41)
Each function 119891 isin L1 is identified with the Boehmian [119891 lowast120575119899120575119899] Also [119891
119899120575119899] lowast 120575119899= 119891119899isin L1 for every 119899 isin N
Since [120575119899120575119899] corresponds to Dirac delta distribution 120575 the
119896th-derivative of each 120588 isin 120588L1 is defined as
D119896
120588 = 120588 lowastD119896
120575 (42)
The integral of a Boehmian 120588 = [119891119899120575119899] isin 120588L1 is defined as
[11]
int
infin
minusinfin
120588 (119909) 119889119909 = int
infin
minusinfin
1198911(119909) 119889119909 (43)
It is of great interest to observe the following example
Example 9 Every infinitely smooth function119891(119909) isinL1 suchthat D119896119891(119909) notin L1 is integrable as Boehmian but not inte-grable as function
The following has importance in the sense of analysis
Theorem 10 Let [119891119899120575119899] isin 120588L1 then the sequence
BA(119891119899) (119910)
= int
infin
minusinfin
(A119900
119891119899(119909) cos (119909119910) +A119890119891
119899(119909) sin (119909119910)) 119889119909
(44)
converges uniformly on each compact subsetK ofR
Proof By aid of theTheorem 8 and the concept of quotient ofsequences we have
BA(119891119899) (119910) =B
A(119891119899lowast120575119896
120575119896
) (119910)
=BA(119891119899lowast 120575119896
120575119896
) (119910)
=BA(119891119896
120575119896
lowast 120575119899) (119910)
997888rarrBA119891119896
120575119896
(119910) as 119899 997888rarr infin
(45)
where convergence ranges over compact subsets of R Thetheorem is completely proved
Let [119891119899120575119899] isin 120588L1 then by virtue ofTheorem 10 we define
the Hartley-Hilbert transform of the Lebesgue Boehmian[119891119899120575119899] as
B
A[119891119899
120575119899
] = lim119899rarrinfin
BA119891119899 (46)
on compact subsets ofRThe next objective is to establish that our definition is well
defined Let [119891119899120575119899] = [119892
119899120574119899] in 120588L1 then
119891119899lowast 120574119898= 119892119898lowast 120575119899 for every 119898 119899 isinN (47)
Hence applying the Hartley-Hilbert transform to both sidesof the above equation and using the concept of quotients ofsequences imply
BA(119891119899lowast 120574119898) =B
A(119892119898lowast 120575119899) =B
A(119892119899lowast 120575119898) (48)
Thus in particular for 119899 = 119898 and considering Theorem 3we get
lim119899rarrinfin
BA119891119899= lim119899rarrinfin
BA119892119899 (49)
Hence
B
A[119891119899
120575119899
] =B
A[119892119899
120574119899
] (50)
Definition (46) is therefore well defined
Theorem 11 The generalized transformBA is linear
Abstract and Applied Analysis 5
Proof Let 1205881= [119891119899120575119899] and 120588
2= [119892119899120574119899] be arbitrary in 120588L1
and 120572 isin C then 1205881+1205882= [(119891119899lowast120574119899+119892119899lowast120575119899)120575119899lowast120574119899] Hence
employing (46) yields
B
A(1205881+ 1205882) = lim119899rarrinfin
(BA(119891119899lowast 120574119899) +B
A(119892119899lowast 120575119899))
(51)
ByTheorem 8 we get
B
A(1205881+ 1205882) = lim119899rarrinfin
BA119891119899+ lim119899rarrinfin
BA119892119899 (52)
Hence
B
A(1205881+ 1205882) =B
A1205881+B
A1205882 (53)
Also for each complex number 120572 we have
B
A(1205721205881) =B
A[120572119891119899
120575119899
]
= 120572 lim119899rarrinfin
BA119891119899
= 120572B
A1205881
(54)
Hence we have the following theorem
Theorem 12 Let 120588 isin 120588L1 and (120598119899) isin Δ then
B
A(120588 lowast 120598
119899) =B
A120588 =B
A(120598119899lowast 120588) (55)
Proof Let 120588 = [119891119899120575119899] isin 120588L1 then
BA(120588 lowast 120598
119899) =BA[119891119899lowast
120598119899120575119899] = lim
119899rarrinfinBA(119891119899lowast 120598119899)
HenceBA(120588 lowast 120598
119899) = lim
119899rarrinfinBA119891119899=BA120588
Similarly we proceed forBA120588 =BA(120598119899lowast 120588)
This completes the theorem The following theorem isobvious
Theorem 13 IfBA1205881= 0 then 120588
1= 0
Theorem14 TheHartley-Hilbert transformBA is continuouswith respect to the 120575-convergence
Proof Let 120588119899
120575
997888rarr 120588 in 120588L1 as 119899 rarr infin then we show thatBA120588119899
120575
997888rarrBA120588 as 119899 rarr infin Using ([11Theorem26]) we find
119891119899119896 119891119896isinL1 (120575
119896) isin Δ such that [119891
119899119896120575119896] = 120588119899 [119891119896120575119896] = 120588
and 119891119899119896rarr 119891119896as 119899 rarr infin 119896 isinN
Applying the Hartley-Hilbert transform for both sidesimplies BA
119891119899119896rarr BA
119891119896in the space of continuous
functions Therefore considering limits we get
B
A[119891119899119896
120575119896
] 997888rarrB
A[119891119896
120575119896
] (56)
This completes the proof of the theorem
Theorem15 TheHartley-Hilbert transformBA is continuouswith respect to the Δ-convergence
Proof Let 120588119899
Δ
997888rarr 120588 as 119899 rarr infin in 120588L1 then there is 119891119899isin L1
and 120575119899isin Δ such that
(120588119899minus 120588) lowast 120575
119899= [119891119899lowast 120575119899
120575119896
] 119891119899997888rarr 0 as 119899 997888rarr infin
(57)
Thus by the aid of Theorem 3 and the hypothesis of thetheorem we have
B
A((120588119899minus 120588) lowast 120575
119899) =B
A[119891119899lowast 120575119899
120575119896
]
997888rarrBA(119891119899lowast 120575119899) as 119899 997888rarr infin
997888rarrBA119891119899
as 119899 997888rarr infin
997888rarr 0 as 119899 997888rarr infin(58)
ThereforeBA(120588119899minus 120588) rarr 0 as 119899 rarr infin ThusBA
120588119899
Δ
997888rarr
BA120588 as 119899 rarr infinThis completes the proof
Lemma 16 Let [119891119899120575119899] isin 120588L1 and 120575 has the usual meaning
of (34) then
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899
120575119899
] (59)
Proof Let 120588 = [119891119899120575119899] isin 120588L1 then
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899lowast 120575
120575119899
]
= lim119899rarrinfin
BA(119891119899lowast 120575)
= lim119899rarrinfin
BA119891119899
(60)
Hence
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899
120575119899
] (61)
Theorem 17 The Hartley-Hilbert transform BA is one-to-one
Proof LetBA[119891119899120575119899] =BA[119892119899120574119899] then by the aid of (46)
we get
lim119899rarrinfin
BA119891119899= lim119899rarrinfin
BA119892119899 (62)
Hence
BA( lim119899rarrinfin
119891119899) =B
A( lim119899rarrinfin
119892119899) (63)
that isBA119891 =BA
119892 The fact thatBA is one-to-one implies119891 = 119892 Hence we have the following theorem
6 Abstract and Applied Analysis
4 A Comparative Study Fourier-HilbertTransform
In [6 7] the Hilbert transform via the Fourier transform of119891(119909) is defined as
BF(119891) (119910)
=1
120587int
infin
0
(F119894(119891) (119909) cos (119909119910) minusF
119903(119891) (119909) sin (119909119910)) 119889119909
(64)
where
F119903(119891) (119909) = int
infin
0
119891 (119905) cos (119909119905) 119889119905
F119894(119891) (119909) = int
infin
0
119891 (119905) sin (119909119905) 119889119905(65)
are respectively the real and imaginary components of theFourier transform of 119891 which are related by
F (119891) (119909) = F119903(119891) (119909) minus 119894F
119894(119891) (119909) (66)
It is interesting to know that a concrete relationship betweenF119903A119890 and F
119894A119900 is described as F
119903(119909) = A119890(119909) and
F119894(119909) = A119900(119909) [2] Those equations above justify the
following statements of the next theorems
Theorem 18 Let [119891119899120575119899] isin 120588L1 then the sequence of Fourier-
Hilbert transforms of (119891119899) satisfies
BF(119891119899) (119910)
=1
120587int
infin
0
(F119894(119891119899) (119909) cos (119909119910) minusF
119903(119891119899) (119909) sin (119909119910)) 119889119909
(67)
and converges uniformly on each compact subsetK ofR
Thus for [119891119899120575119899] isin 120588L1 the Fourier-Hilbert transform of
[119891119899120575119899] is similarly defined by
B
F[119891119899
120575119899
] = lim119899rarrinfin
BF119891119899 (68)
on compact subsets ofRThe following theorems are stated and their proofs are
justified for similar reasons We prefer to we omit the details
Theorem 19 The generalized Fourier-Hilbert transformBF
is linear
Theorem 20 BF(120588 lowast 120598
119899) =BF120588 =BF(120598 lowast 120588) (120598
119899) isin Δ
Theorem 21 IfBF1205881= 0 then 120588
1= 0
Theorem 22 If 120588119899
Δ
997888rarr 120588 as 119899 rarr infin in 120588L1 thenBF120588119899
Δ
997888rarr
BF120588 as 119899 rarr infin in 120588L1 on compact subsets
Theorem 23 The Fourier-Hilbert transformBF is continu-ous with respect to the 120575-convergence
Theorem 24 The Fourier-Hilbert transformBF is continu-ous with respect to the Δ-convergence
Proofs of the above theorems are similar to that given forthe corresponding ones of Hartley-Hilbert transform
References
[1] A H Zemanian Generalized Integral Transformations DoverPublications New York NY USA 2nd edition 1987
[2] R S Pathak Integral Transforms of Generalized Functions andTheir Applications Gordon and Breach Science PublishersNorth Vancouver Canada 1997
[3] S K Q Al-Omari D Loonker P K Banerji and S L KallaldquoFourier sine (cosine) transform for ultradistributions and theirextensions to tempered and ultraBoehmian spacesrdquo IntegralTransforms and Special Functions vol 19 no 5-6 pp 453ndash4622008
[4] S Al-Omari and A Kılıcman ldquoOn the generalized Hartley-Hilbert and fourier-Hilbert transformsrdquo Advances in DifferenceEquations vol 2012 p 232 2012
[5] R P Millane ldquoAnalytic properties of the Hartley transform andtheir applicationsrdquo Proceedings of the IEEE vol 82 no 3 pp413ndash428 1994
[6] N Sundararajan and Y Srinivas ldquoFourier-Hilbert versusHartley-Hilbert transforms with some geophysical applica-tionsrdquo Journal of Applied Geophysics vol 71 no 4 pp 157ndash1612010
[7] N Sundarajan ldquoFourier and hartley transforms amathematicaltwinrdquo Indian Journal of Pure and Applied Mathematics vol 8no 10 pp 1361ndash1365 1997
[8] T K Boehme ldquoThe support of Mikusinski operatorsrdquo Transac-tions of theAmericanMathematical Society vol 176 pp 319ndash3341973
[9] P Mikusinski ldquoFourier transform for integrable BoehmiansrdquoThe Rocky Mountain Journal of Mathematics vol 17 no 3 pp577ndash582 1987
[10] S K Q Al-Omari and A Kılıcman ldquoOn diffraction Fresneltransforms for Boehmiansrdquo Abstract and Applied Analysis vol2011 Article ID 712746 11 pages 2011
[11] P Mikusinski ldquoTempered Boehmians and ultradistributionsrdquoProceedings of the American Mathematical Society vol 123 no3 pp 813ndash817 1995
[12] S K Q Al-Omari and A Kılıcman ldquoNote on Boehmians forclass of optical Fresnel wavelet transformsrdquo Journal of FunctionSpaces and Applications vol 2012 Article ID 405368 14 pages2012
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2 Abstract and Applied Analysis
or
119891 (119909) = int
infin
minusinfin
A (119891) cas (2120587119891119909) 119889119891 (7)
The theory of convolutions of integral transforms has beendeveloped for a long time and is applied in many fields ofmathematics Historically the convolution product [2]
(119891 lowast 119892) (119910) = int
infin
minusinfin
119891 (119909) 119892 (119909 minus 119910) 119889119910 (8)
has a relationship with the Fourier transform with the factor-ization property
F (119891 lowast 119892) (119910) = F (119891) (119910)F (119892) (119910) (9)
Themore complicated convolution theorem of Hartley trans-forms compared to that of Fourier transforms is that
A (119891 lowast 119892) (119910) =1
2G (A119891 timesA119892) (119910) (10)
where
G (119891 lowast 119892) (119910) = 119891 (119910) 119892 (119910) + 119891 (119910) 119892 (minus119910)
+ 119891 (119910) 119892 (119910) minus 119891 (minus119910) 119892 (minus119910)
(11)
Some properties of Hartley transforms can be listed asfollows
(i) Linearity if 119891 and 119892 are real functions then
A (119886119891 + 119887119892) (119910) = 119886A (119891) (119910) + 119887A (119892) (119910) 119886 119887 isinR
(12)
(ii) Scaling if 119891 is a real function then
int
infin
minusinfin
119891 (120572120589) cas (2120587119910120589) 119889120589 = 1119886(A119891) (
119910
119886) (13)
2 Distributional Hartley-Hilbert Transformof Compact Support
TheHilbert transform via the Hartley transform is defined by[6 7]
BA(119910) = minus
1
120587int
infin
0
(A119900
(119909) cos (119909119910) +A119890 (119909) sin (119909119910)) 119889119909
(14)
where
A119900
(119909) =A (119909) minusA (minus119909)
2
A119890
(119909) =A (119909) +A (minus119909)
2
(15)
are the respective odd and even components of (2)We denote C(R) C(R) = C the space of smooth
functions and C1015840(R) C1015840(R) = C1015840 the strong dual of C ofdistributions of compact support overR
Following is the convolution theorem ofBA
Theorem 1 (ConvolutionTheorem) Let 119891 and 119892 isin C then
BA(119891 lowast 119892) (119910) = int
infin
0
(1198961(119909) cos (119910119909) + 119896
2(119909) sin (119910119909)) 119889119909
(16)
where
1198961(119909) = A
119890
119891 (119909)A119900
119892 (119909) +A119900
119891 (119909)A119890
119892 (119909)
1198962(119909) = A
119890
119891 (119909)A119890
119892 (119909) minusA119900
119891 (119909)A119900
119892 (119909)
(17)
Proof To prove this theorem it is sufficient to establish that
1198961(119909) = A
119900
(119891 lowast 119892) (119909) (18)
1198962(119909) = A
119890
(119891 lowast 119892) (119909) (19)
Therefore we have
A119900
(119891 lowast 119892) (119909)
= int
infin
minusinfin
(int
infin
minusinfin
119891 (120574) 119892 (119910 minus 120574) 119889119903) cas (119909119910) 119889119910
= int
infin
minusinfin
119891 (120574)int
infin
minusinfin
119892 (119910 minus 120574) cas (119909119910) 119889119910119889119903
(20)
The substitution 119910 minus 120574 = 119911 and using of (1) together withFubini theorem imply
A119900
(119891 lowast 119892) (119909)
= int
infin
minusinfin
119891 (120574)int
infin
minusinfin
119892 (119911) (cos (119909 (119911 + 120574)) + sin (119909 (119911 + 120574)))
times 119889119911119889119903
(21)
By invoking the formulae
cos (119909 (119911 + 120574)) = cos (119909119911) cos (119909120574) minus sin (119909119911) sin (119909120574)
sin (119909 (119911 + 120574)) = sin (119909119911) cos (119909120574) + cos (119909119911) sin (119909120574) (22)
then (18) follows from simple computation Proof of (19) hasa similar techniqueHence the theorem is completely proved
It is of interest to know that cos(119909119910) and sin(119909119910) aremembers of C and therefore A119900119891A119890119891 isin C1015840 This leads tothe following statement
Definition 2 Let 119891 isin C1015840 then we define the distributionalHartley-Hilbert transform of 119891 as
B
A119891 (119910) = ⟨A
119900
119891 (119909) cos (119909119910)⟩ + ⟨A119890119891 (119909) sin (119909119910)⟩ (23)
The extended transformBA119891 is clearly well defined for
each 119891 isin C1015840
Abstract and Applied Analysis 3
Theorem 3 The distributional Hartley-Hilbert transformBA119891 is linear
Proof Let 119891 119892 isin C1015840 then their components A119890119891A119900119891A119890119892A119900119892 isin C1015840 HenceB
A(119891 + 119892) (119910) = ⟨A
119900
(119891 + 119892) (119909) cos (119909119910)⟩
+ ⟨A119890
(119891 + 119892) (119909) sin (119909119910)⟩ (24)
By factoring and rearranging components we get thatB
A(119891 + 119892) (119910) =
B
A119891 (119910) +
B
A119892 (119910) (25)
FurthermoreB
A(119896119891) (119910) = ⟨119896A
119900
119891 (119909) cos (119909119910)⟩
+ ⟨119896A119890
119891 (119909) sin (119909119910)⟩ (26)
HenceB
A(119896119891) (119910) = 119896
B
A119891 (119910) (27)
This completes the proof of the theorem
Theorem 4 Let 119891 isin C1015840 thenBA119891 is a continuous mapping
onC1015840
Proof Let119891119899 119891 isin C1015840 119899 isinN and119891
119899rarr 119891 as 119899 rarr infinThen
B
A119891119899(119910) = ⟨A
119900
119891119899(119909) cos (119909119910)⟩ + ⟨A119890119891
119899(119909) sin (119909119910)⟩
997888rarr ⟨A119900
119891 (119909) cos (119909119910)⟩ + ⟨A119890119891 (119909) sin (119909119910)⟩
=B
A119891 (119910) as 119899 997888rarr infin
(28)
Hence we have the following theorem
Theorem 5 The mappingBA119891 is one-to-one
Proof Let 119891 119892 isin C1015840 and thatBA119891 =BA119892 then using (23)
we get
⟨A119900
119891 (119909) cos119909119910⟩ + ⟨A119890119891 (119909) sin119909119910⟩
= ⟨A119900
119892 (119909) cos119909119910⟩ + ⟨A119890119892 (119909) sin119909119910⟩ (29)
Basic properties of inner product implies
⟨A119900
119891 (119909) minusA119900
119892 (119909) cos (119909119910)⟩
+ ⟨A119890
119891 (119909) minusA119890
119892 (119909) sin (119909119910)⟩ = 0(30)
Hence
A119900
119891 (119909) = A119900
119892 (119909) A119890
119891 (119909) = A119890
119892 (119909) (31)
ThereforeA119891 (119909) = A
119900
119891 (119909) +A119890
119891 (119909)
= A119900
119892 (119909) +A119890
119892 (119909) = A119892 (119909)(32)
for all 119909 This completes the proof of the theorem
Theorem 6 Let 119891 isin C1015840 then 119891 is analytic and
D119896
119910
B
A119891 (119910) = ⟨A
119900
119891 (119909) D119896
119910cos (119909119910)⟩
+ ⟨A119890
119891 (119909) D119896
119910sin (119909119910)⟩
(33)
Proof of this theorem is analogous to that of the previoustheorem and is thus avoided
Denote by 120575 the dirac delta function then it is easy to seethat
A119890
120575 (119910) = 1 A119890
120575 (119910) = 0 (34)
3 Lebesgue Space of Boehmians forHartley-Hilbert Transforms
The original construction of Boehmians produce a concretespace of generalized functions Since the space of Boehmianswas introduced many spaces of Boehmians were defined Inreferences we list selected papers introducing different spacesof Boehmians One of the main motivations for introducingdifferent spaces of Boehmians was the generalization ofintegral transforms The idea requires a proper choice of aspace of functions for which a given integral transform iswell defined a choice of a class of delta sequences that istransformed by that integral transform to a well-behavedclass of approximate identities and finally a convolutionproduct that behaves well under the transform If theseconditions are met the transform has usually an extensionto the constructed space of Boehmians and the extension hasdesirable properties For general construction of Boehmianssee [8ndash12]
LetD be the space of test functions of bounded supportBy delta sequence we mean a subset of D of sequences (120575
119899)
such that
int
infin
minusinfin
120575119899(119909) 119889119909 = 1
10038171003817100381710038171205751198991003817100381710038171003817 = int
infin
minusinfin
1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 ltM 0 ltM isinR
lim119899rarrinfin
int|119909|gt120576
1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 = 0 for each 120576 gt 0
(35)
where 120576(120575119899)(119909) = 119909 isinR 120575
119899(119909) = 0
The set of all such delta sequences is usually denoted asΔEach element in Δ corresponds to the dirac delta function 120575for large values of 119899
Proposition 7 Let (120575119899) isin Δ then
A119890
120575119899(119910) = int
infin
minusinfin
120575119899(119909) cos (119909119910) 119889119909 997888rarr 1 as 119899 997888rarr infin
A119900
120575119899(119910) = int
infin
minusinfin
120575119899(119909) sin (119909119910) 119889119909 997888rarr 0 as 119899 997888rarr infin
(36)
Let L1(R) L1(R) = L1 be the space of complexvalued Lebesgue integrable functions From Proposition 7 weestablish the following theorem
4 Abstract and Applied Analysis
Theorem 8 Let 119891 isin L1 then BA(119891 lowast 120575
119899)(119910) rarr BA
119891(119910)
as 119899 rarr infin
Proof For 119891 isinL1 (120575119899) isin Δ then using of (14) implies
BA(119891 lowast 120575
119899) (119910)
= int
infin
minusinfin
(A119900
(119891 lowast 120575119899) (119909) cos119909119910 +A119890 (119891 lowast 120575
119899) (119909) sin119909119910) 119889119909
(37)
Since
(119891 lowast 120575119899) (120577) = int
infin
minusinfin
119891 (119905) 120575119899(120577 minus 119905) 119889119905 997888rarr 119891 (120577) (38)
as 119899 rarr infin we see that
A119900
(119891 lowast 120575119899) (119909) = int
infin
minusinfin
(119891 lowast 120575119899) (120577) sin119909120577119889120577
= int
infin
minusinfin
119891 (119905) int
infin
minusinfin
120575119899(120577 minus 119905) sin (119909120577) 119889120577119889119905
997888rarr int
infin
minusinfin
119891 (119905) sin (119909119905) 119889119905
= A119900
(119891) (119909)
(39)
Similarly
A119890
(119891 lowast 120575119899) (119909) 997888rarr A
119890
(119891) (119909) as 119899 997888rarr infin (40)
Therefore invoking the above equations in (37) we getBA(119891 lowast 120575
119899)(119910) rarr BA
119891(119910) as 119899 rarr infin Hence thetheorem
Denote by 120588L1 the space of integrable Boehmians then120588L1 is a convolution algebra when multiplication by scalaraddition and convolution is defined as [9]
119896 [119891119899
120575119899
] = [119896119891119899
120575119899
]
[119891119899
120575119899
] + [119892119899
120574119899
] = [119891119899lowast 120574119899+ 119892119899lowast 120575119899
120575119899lowast 120574119899
]
[119891119899
120575119899
] lowast [119892119899
120574119899
] = [119891119899lowast 119892119899
120575119899lowast 120574119899
]
(41)
Each function 119891 isin L1 is identified with the Boehmian [119891 lowast120575119899120575119899] Also [119891
119899120575119899] lowast 120575119899= 119891119899isin L1 for every 119899 isin N
Since [120575119899120575119899] corresponds to Dirac delta distribution 120575 the
119896th-derivative of each 120588 isin 120588L1 is defined as
D119896
120588 = 120588 lowastD119896
120575 (42)
The integral of a Boehmian 120588 = [119891119899120575119899] isin 120588L1 is defined as
[11]
int
infin
minusinfin
120588 (119909) 119889119909 = int
infin
minusinfin
1198911(119909) 119889119909 (43)
It is of great interest to observe the following example
Example 9 Every infinitely smooth function119891(119909) isinL1 suchthat D119896119891(119909) notin L1 is integrable as Boehmian but not inte-grable as function
The following has importance in the sense of analysis
Theorem 10 Let [119891119899120575119899] isin 120588L1 then the sequence
BA(119891119899) (119910)
= int
infin
minusinfin
(A119900
119891119899(119909) cos (119909119910) +A119890119891
119899(119909) sin (119909119910)) 119889119909
(44)
converges uniformly on each compact subsetK ofR
Proof By aid of theTheorem 8 and the concept of quotient ofsequences we have
BA(119891119899) (119910) =B
A(119891119899lowast120575119896
120575119896
) (119910)
=BA(119891119899lowast 120575119896
120575119896
) (119910)
=BA(119891119896
120575119896
lowast 120575119899) (119910)
997888rarrBA119891119896
120575119896
(119910) as 119899 997888rarr infin
(45)
where convergence ranges over compact subsets of R Thetheorem is completely proved
Let [119891119899120575119899] isin 120588L1 then by virtue ofTheorem 10 we define
the Hartley-Hilbert transform of the Lebesgue Boehmian[119891119899120575119899] as
B
A[119891119899
120575119899
] = lim119899rarrinfin
BA119891119899 (46)
on compact subsets ofRThe next objective is to establish that our definition is well
defined Let [119891119899120575119899] = [119892
119899120574119899] in 120588L1 then
119891119899lowast 120574119898= 119892119898lowast 120575119899 for every 119898 119899 isinN (47)
Hence applying the Hartley-Hilbert transform to both sidesof the above equation and using the concept of quotients ofsequences imply
BA(119891119899lowast 120574119898) =B
A(119892119898lowast 120575119899) =B
A(119892119899lowast 120575119898) (48)
Thus in particular for 119899 = 119898 and considering Theorem 3we get
lim119899rarrinfin
BA119891119899= lim119899rarrinfin
BA119892119899 (49)
Hence
B
A[119891119899
120575119899
] =B
A[119892119899
120574119899
] (50)
Definition (46) is therefore well defined
Theorem 11 The generalized transformBA is linear
Abstract and Applied Analysis 5
Proof Let 1205881= [119891119899120575119899] and 120588
2= [119892119899120574119899] be arbitrary in 120588L1
and 120572 isin C then 1205881+1205882= [(119891119899lowast120574119899+119892119899lowast120575119899)120575119899lowast120574119899] Hence
employing (46) yields
B
A(1205881+ 1205882) = lim119899rarrinfin
(BA(119891119899lowast 120574119899) +B
A(119892119899lowast 120575119899))
(51)
ByTheorem 8 we get
B
A(1205881+ 1205882) = lim119899rarrinfin
BA119891119899+ lim119899rarrinfin
BA119892119899 (52)
Hence
B
A(1205881+ 1205882) =B
A1205881+B
A1205882 (53)
Also for each complex number 120572 we have
B
A(1205721205881) =B
A[120572119891119899
120575119899
]
= 120572 lim119899rarrinfin
BA119891119899
= 120572B
A1205881
(54)
Hence we have the following theorem
Theorem 12 Let 120588 isin 120588L1 and (120598119899) isin Δ then
B
A(120588 lowast 120598
119899) =B
A120588 =B
A(120598119899lowast 120588) (55)
Proof Let 120588 = [119891119899120575119899] isin 120588L1 then
BA(120588 lowast 120598
119899) =BA[119891119899lowast
120598119899120575119899] = lim
119899rarrinfinBA(119891119899lowast 120598119899)
HenceBA(120588 lowast 120598
119899) = lim
119899rarrinfinBA119891119899=BA120588
Similarly we proceed forBA120588 =BA(120598119899lowast 120588)
This completes the theorem The following theorem isobvious
Theorem 13 IfBA1205881= 0 then 120588
1= 0
Theorem14 TheHartley-Hilbert transformBA is continuouswith respect to the 120575-convergence
Proof Let 120588119899
120575
997888rarr 120588 in 120588L1 as 119899 rarr infin then we show thatBA120588119899
120575
997888rarrBA120588 as 119899 rarr infin Using ([11Theorem26]) we find
119891119899119896 119891119896isinL1 (120575
119896) isin Δ such that [119891
119899119896120575119896] = 120588119899 [119891119896120575119896] = 120588
and 119891119899119896rarr 119891119896as 119899 rarr infin 119896 isinN
Applying the Hartley-Hilbert transform for both sidesimplies BA
119891119899119896rarr BA
119891119896in the space of continuous
functions Therefore considering limits we get
B
A[119891119899119896
120575119896
] 997888rarrB
A[119891119896
120575119896
] (56)
This completes the proof of the theorem
Theorem15 TheHartley-Hilbert transformBA is continuouswith respect to the Δ-convergence
Proof Let 120588119899
Δ
997888rarr 120588 as 119899 rarr infin in 120588L1 then there is 119891119899isin L1
and 120575119899isin Δ such that
(120588119899minus 120588) lowast 120575
119899= [119891119899lowast 120575119899
120575119896
] 119891119899997888rarr 0 as 119899 997888rarr infin
(57)
Thus by the aid of Theorem 3 and the hypothesis of thetheorem we have
B
A((120588119899minus 120588) lowast 120575
119899) =B
A[119891119899lowast 120575119899
120575119896
]
997888rarrBA(119891119899lowast 120575119899) as 119899 997888rarr infin
997888rarrBA119891119899
as 119899 997888rarr infin
997888rarr 0 as 119899 997888rarr infin(58)
ThereforeBA(120588119899minus 120588) rarr 0 as 119899 rarr infin ThusBA
120588119899
Δ
997888rarr
BA120588 as 119899 rarr infinThis completes the proof
Lemma 16 Let [119891119899120575119899] isin 120588L1 and 120575 has the usual meaning
of (34) then
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899
120575119899
] (59)
Proof Let 120588 = [119891119899120575119899] isin 120588L1 then
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899lowast 120575
120575119899
]
= lim119899rarrinfin
BA(119891119899lowast 120575)
= lim119899rarrinfin
BA119891119899
(60)
Hence
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899
120575119899
] (61)
Theorem 17 The Hartley-Hilbert transform BA is one-to-one
Proof LetBA[119891119899120575119899] =BA[119892119899120574119899] then by the aid of (46)
we get
lim119899rarrinfin
BA119891119899= lim119899rarrinfin
BA119892119899 (62)
Hence
BA( lim119899rarrinfin
119891119899) =B
A( lim119899rarrinfin
119892119899) (63)
that isBA119891 =BA
119892 The fact thatBA is one-to-one implies119891 = 119892 Hence we have the following theorem
6 Abstract and Applied Analysis
4 A Comparative Study Fourier-HilbertTransform
In [6 7] the Hilbert transform via the Fourier transform of119891(119909) is defined as
BF(119891) (119910)
=1
120587int
infin
0
(F119894(119891) (119909) cos (119909119910) minusF
119903(119891) (119909) sin (119909119910)) 119889119909
(64)
where
F119903(119891) (119909) = int
infin
0
119891 (119905) cos (119909119905) 119889119905
F119894(119891) (119909) = int
infin
0
119891 (119905) sin (119909119905) 119889119905(65)
are respectively the real and imaginary components of theFourier transform of 119891 which are related by
F (119891) (119909) = F119903(119891) (119909) minus 119894F
119894(119891) (119909) (66)
It is interesting to know that a concrete relationship betweenF119903A119890 and F
119894A119900 is described as F
119903(119909) = A119890(119909) and
F119894(119909) = A119900(119909) [2] Those equations above justify the
following statements of the next theorems
Theorem 18 Let [119891119899120575119899] isin 120588L1 then the sequence of Fourier-
Hilbert transforms of (119891119899) satisfies
BF(119891119899) (119910)
=1
120587int
infin
0
(F119894(119891119899) (119909) cos (119909119910) minusF
119903(119891119899) (119909) sin (119909119910)) 119889119909
(67)
and converges uniformly on each compact subsetK ofR
Thus for [119891119899120575119899] isin 120588L1 the Fourier-Hilbert transform of
[119891119899120575119899] is similarly defined by
B
F[119891119899
120575119899
] = lim119899rarrinfin
BF119891119899 (68)
on compact subsets ofRThe following theorems are stated and their proofs are
justified for similar reasons We prefer to we omit the details
Theorem 19 The generalized Fourier-Hilbert transformBF
is linear
Theorem 20 BF(120588 lowast 120598
119899) =BF120588 =BF(120598 lowast 120588) (120598
119899) isin Δ
Theorem 21 IfBF1205881= 0 then 120588
1= 0
Theorem 22 If 120588119899
Δ
997888rarr 120588 as 119899 rarr infin in 120588L1 thenBF120588119899
Δ
997888rarr
BF120588 as 119899 rarr infin in 120588L1 on compact subsets
Theorem 23 The Fourier-Hilbert transformBF is continu-ous with respect to the 120575-convergence
Theorem 24 The Fourier-Hilbert transformBF is continu-ous with respect to the Δ-convergence
Proofs of the above theorems are similar to that given forthe corresponding ones of Hartley-Hilbert transform
References
[1] A H Zemanian Generalized Integral Transformations DoverPublications New York NY USA 2nd edition 1987
[2] R S Pathak Integral Transforms of Generalized Functions andTheir Applications Gordon and Breach Science PublishersNorth Vancouver Canada 1997
[3] S K Q Al-Omari D Loonker P K Banerji and S L KallaldquoFourier sine (cosine) transform for ultradistributions and theirextensions to tempered and ultraBoehmian spacesrdquo IntegralTransforms and Special Functions vol 19 no 5-6 pp 453ndash4622008
[4] S Al-Omari and A Kılıcman ldquoOn the generalized Hartley-Hilbert and fourier-Hilbert transformsrdquo Advances in DifferenceEquations vol 2012 p 232 2012
[5] R P Millane ldquoAnalytic properties of the Hartley transform andtheir applicationsrdquo Proceedings of the IEEE vol 82 no 3 pp413ndash428 1994
[6] N Sundararajan and Y Srinivas ldquoFourier-Hilbert versusHartley-Hilbert transforms with some geophysical applica-tionsrdquo Journal of Applied Geophysics vol 71 no 4 pp 157ndash1612010
[7] N Sundarajan ldquoFourier and hartley transforms amathematicaltwinrdquo Indian Journal of Pure and Applied Mathematics vol 8no 10 pp 1361ndash1365 1997
[8] T K Boehme ldquoThe support of Mikusinski operatorsrdquo Transac-tions of theAmericanMathematical Society vol 176 pp 319ndash3341973
[9] P Mikusinski ldquoFourier transform for integrable BoehmiansrdquoThe Rocky Mountain Journal of Mathematics vol 17 no 3 pp577ndash582 1987
[10] S K Q Al-Omari and A Kılıcman ldquoOn diffraction Fresneltransforms for Boehmiansrdquo Abstract and Applied Analysis vol2011 Article ID 712746 11 pages 2011
[11] P Mikusinski ldquoTempered Boehmians and ultradistributionsrdquoProceedings of the American Mathematical Society vol 123 no3 pp 813ndash817 1995
[12] S K Q Al-Omari and A Kılıcman ldquoNote on Boehmians forclass of optical Fresnel wavelet transformsrdquo Journal of FunctionSpaces and Applications vol 2012 Article ID 405368 14 pages2012
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Theorem 3 The distributional Hartley-Hilbert transformBA119891 is linear
Proof Let 119891 119892 isin C1015840 then their components A119890119891A119900119891A119890119892A119900119892 isin C1015840 HenceB
A(119891 + 119892) (119910) = ⟨A
119900
(119891 + 119892) (119909) cos (119909119910)⟩
+ ⟨A119890
(119891 + 119892) (119909) sin (119909119910)⟩ (24)
By factoring and rearranging components we get thatB
A(119891 + 119892) (119910) =
B
A119891 (119910) +
B
A119892 (119910) (25)
FurthermoreB
A(119896119891) (119910) = ⟨119896A
119900
119891 (119909) cos (119909119910)⟩
+ ⟨119896A119890
119891 (119909) sin (119909119910)⟩ (26)
HenceB
A(119896119891) (119910) = 119896
B
A119891 (119910) (27)
This completes the proof of the theorem
Theorem 4 Let 119891 isin C1015840 thenBA119891 is a continuous mapping
onC1015840
Proof Let119891119899 119891 isin C1015840 119899 isinN and119891
119899rarr 119891 as 119899 rarr infinThen
B
A119891119899(119910) = ⟨A
119900
119891119899(119909) cos (119909119910)⟩ + ⟨A119890119891
119899(119909) sin (119909119910)⟩
997888rarr ⟨A119900
119891 (119909) cos (119909119910)⟩ + ⟨A119890119891 (119909) sin (119909119910)⟩
=B
A119891 (119910) as 119899 997888rarr infin
(28)
Hence we have the following theorem
Theorem 5 The mappingBA119891 is one-to-one
Proof Let 119891 119892 isin C1015840 and thatBA119891 =BA119892 then using (23)
we get
⟨A119900
119891 (119909) cos119909119910⟩ + ⟨A119890119891 (119909) sin119909119910⟩
= ⟨A119900
119892 (119909) cos119909119910⟩ + ⟨A119890119892 (119909) sin119909119910⟩ (29)
Basic properties of inner product implies
⟨A119900
119891 (119909) minusA119900
119892 (119909) cos (119909119910)⟩
+ ⟨A119890
119891 (119909) minusA119890
119892 (119909) sin (119909119910)⟩ = 0(30)
Hence
A119900
119891 (119909) = A119900
119892 (119909) A119890
119891 (119909) = A119890
119892 (119909) (31)
ThereforeA119891 (119909) = A
119900
119891 (119909) +A119890
119891 (119909)
= A119900
119892 (119909) +A119890
119892 (119909) = A119892 (119909)(32)
for all 119909 This completes the proof of the theorem
Theorem 6 Let 119891 isin C1015840 then 119891 is analytic and
D119896
119910
B
A119891 (119910) = ⟨A
119900
119891 (119909) D119896
119910cos (119909119910)⟩
+ ⟨A119890
119891 (119909) D119896
119910sin (119909119910)⟩
(33)
Proof of this theorem is analogous to that of the previoustheorem and is thus avoided
Denote by 120575 the dirac delta function then it is easy to seethat
A119890
120575 (119910) = 1 A119890
120575 (119910) = 0 (34)
3 Lebesgue Space of Boehmians forHartley-Hilbert Transforms
The original construction of Boehmians produce a concretespace of generalized functions Since the space of Boehmianswas introduced many spaces of Boehmians were defined Inreferences we list selected papers introducing different spacesof Boehmians One of the main motivations for introducingdifferent spaces of Boehmians was the generalization ofintegral transforms The idea requires a proper choice of aspace of functions for which a given integral transform iswell defined a choice of a class of delta sequences that istransformed by that integral transform to a well-behavedclass of approximate identities and finally a convolutionproduct that behaves well under the transform If theseconditions are met the transform has usually an extensionto the constructed space of Boehmians and the extension hasdesirable properties For general construction of Boehmianssee [8ndash12]
LetD be the space of test functions of bounded supportBy delta sequence we mean a subset of D of sequences (120575
119899)
such that
int
infin
minusinfin
120575119899(119909) 119889119909 = 1
10038171003817100381710038171205751198991003817100381710038171003817 = int
infin
minusinfin
1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 ltM 0 ltM isinR
lim119899rarrinfin
int|119909|gt120576
1003816100381610038161003816120575119899 (119909)1003816100381610038161003816 119889119909 = 0 for each 120576 gt 0
(35)
where 120576(120575119899)(119909) = 119909 isinR 120575
119899(119909) = 0
The set of all such delta sequences is usually denoted asΔEach element in Δ corresponds to the dirac delta function 120575for large values of 119899
Proposition 7 Let (120575119899) isin Δ then
A119890
120575119899(119910) = int
infin
minusinfin
120575119899(119909) cos (119909119910) 119889119909 997888rarr 1 as 119899 997888rarr infin
A119900
120575119899(119910) = int
infin
minusinfin
120575119899(119909) sin (119909119910) 119889119909 997888rarr 0 as 119899 997888rarr infin
(36)
Let L1(R) L1(R) = L1 be the space of complexvalued Lebesgue integrable functions From Proposition 7 weestablish the following theorem
4 Abstract and Applied Analysis
Theorem 8 Let 119891 isin L1 then BA(119891 lowast 120575
119899)(119910) rarr BA
119891(119910)
as 119899 rarr infin
Proof For 119891 isinL1 (120575119899) isin Δ then using of (14) implies
BA(119891 lowast 120575
119899) (119910)
= int
infin
minusinfin
(A119900
(119891 lowast 120575119899) (119909) cos119909119910 +A119890 (119891 lowast 120575
119899) (119909) sin119909119910) 119889119909
(37)
Since
(119891 lowast 120575119899) (120577) = int
infin
minusinfin
119891 (119905) 120575119899(120577 minus 119905) 119889119905 997888rarr 119891 (120577) (38)
as 119899 rarr infin we see that
A119900
(119891 lowast 120575119899) (119909) = int
infin
minusinfin
(119891 lowast 120575119899) (120577) sin119909120577119889120577
= int
infin
minusinfin
119891 (119905) int
infin
minusinfin
120575119899(120577 minus 119905) sin (119909120577) 119889120577119889119905
997888rarr int
infin
minusinfin
119891 (119905) sin (119909119905) 119889119905
= A119900
(119891) (119909)
(39)
Similarly
A119890
(119891 lowast 120575119899) (119909) 997888rarr A
119890
(119891) (119909) as 119899 997888rarr infin (40)
Therefore invoking the above equations in (37) we getBA(119891 lowast 120575
119899)(119910) rarr BA
119891(119910) as 119899 rarr infin Hence thetheorem
Denote by 120588L1 the space of integrable Boehmians then120588L1 is a convolution algebra when multiplication by scalaraddition and convolution is defined as [9]
119896 [119891119899
120575119899
] = [119896119891119899
120575119899
]
[119891119899
120575119899
] + [119892119899
120574119899
] = [119891119899lowast 120574119899+ 119892119899lowast 120575119899
120575119899lowast 120574119899
]
[119891119899
120575119899
] lowast [119892119899
120574119899
] = [119891119899lowast 119892119899
120575119899lowast 120574119899
]
(41)
Each function 119891 isin L1 is identified with the Boehmian [119891 lowast120575119899120575119899] Also [119891
119899120575119899] lowast 120575119899= 119891119899isin L1 for every 119899 isin N
Since [120575119899120575119899] corresponds to Dirac delta distribution 120575 the
119896th-derivative of each 120588 isin 120588L1 is defined as
D119896
120588 = 120588 lowastD119896
120575 (42)
The integral of a Boehmian 120588 = [119891119899120575119899] isin 120588L1 is defined as
[11]
int
infin
minusinfin
120588 (119909) 119889119909 = int
infin
minusinfin
1198911(119909) 119889119909 (43)
It is of great interest to observe the following example
Example 9 Every infinitely smooth function119891(119909) isinL1 suchthat D119896119891(119909) notin L1 is integrable as Boehmian but not inte-grable as function
The following has importance in the sense of analysis
Theorem 10 Let [119891119899120575119899] isin 120588L1 then the sequence
BA(119891119899) (119910)
= int
infin
minusinfin
(A119900
119891119899(119909) cos (119909119910) +A119890119891
119899(119909) sin (119909119910)) 119889119909
(44)
converges uniformly on each compact subsetK ofR
Proof By aid of theTheorem 8 and the concept of quotient ofsequences we have
BA(119891119899) (119910) =B
A(119891119899lowast120575119896
120575119896
) (119910)
=BA(119891119899lowast 120575119896
120575119896
) (119910)
=BA(119891119896
120575119896
lowast 120575119899) (119910)
997888rarrBA119891119896
120575119896
(119910) as 119899 997888rarr infin
(45)
where convergence ranges over compact subsets of R Thetheorem is completely proved
Let [119891119899120575119899] isin 120588L1 then by virtue ofTheorem 10 we define
the Hartley-Hilbert transform of the Lebesgue Boehmian[119891119899120575119899] as
B
A[119891119899
120575119899
] = lim119899rarrinfin
BA119891119899 (46)
on compact subsets ofRThe next objective is to establish that our definition is well
defined Let [119891119899120575119899] = [119892
119899120574119899] in 120588L1 then
119891119899lowast 120574119898= 119892119898lowast 120575119899 for every 119898 119899 isinN (47)
Hence applying the Hartley-Hilbert transform to both sidesof the above equation and using the concept of quotients ofsequences imply
BA(119891119899lowast 120574119898) =B
A(119892119898lowast 120575119899) =B
A(119892119899lowast 120575119898) (48)
Thus in particular for 119899 = 119898 and considering Theorem 3we get
lim119899rarrinfin
BA119891119899= lim119899rarrinfin
BA119892119899 (49)
Hence
B
A[119891119899
120575119899
] =B
A[119892119899
120574119899
] (50)
Definition (46) is therefore well defined
Theorem 11 The generalized transformBA is linear
Abstract and Applied Analysis 5
Proof Let 1205881= [119891119899120575119899] and 120588
2= [119892119899120574119899] be arbitrary in 120588L1
and 120572 isin C then 1205881+1205882= [(119891119899lowast120574119899+119892119899lowast120575119899)120575119899lowast120574119899] Hence
employing (46) yields
B
A(1205881+ 1205882) = lim119899rarrinfin
(BA(119891119899lowast 120574119899) +B
A(119892119899lowast 120575119899))
(51)
ByTheorem 8 we get
B
A(1205881+ 1205882) = lim119899rarrinfin
BA119891119899+ lim119899rarrinfin
BA119892119899 (52)
Hence
B
A(1205881+ 1205882) =B
A1205881+B
A1205882 (53)
Also for each complex number 120572 we have
B
A(1205721205881) =B
A[120572119891119899
120575119899
]
= 120572 lim119899rarrinfin
BA119891119899
= 120572B
A1205881
(54)
Hence we have the following theorem
Theorem 12 Let 120588 isin 120588L1 and (120598119899) isin Δ then
B
A(120588 lowast 120598
119899) =B
A120588 =B
A(120598119899lowast 120588) (55)
Proof Let 120588 = [119891119899120575119899] isin 120588L1 then
BA(120588 lowast 120598
119899) =BA[119891119899lowast
120598119899120575119899] = lim
119899rarrinfinBA(119891119899lowast 120598119899)
HenceBA(120588 lowast 120598
119899) = lim
119899rarrinfinBA119891119899=BA120588
Similarly we proceed forBA120588 =BA(120598119899lowast 120588)
This completes the theorem The following theorem isobvious
Theorem 13 IfBA1205881= 0 then 120588
1= 0
Theorem14 TheHartley-Hilbert transformBA is continuouswith respect to the 120575-convergence
Proof Let 120588119899
120575
997888rarr 120588 in 120588L1 as 119899 rarr infin then we show thatBA120588119899
120575
997888rarrBA120588 as 119899 rarr infin Using ([11Theorem26]) we find
119891119899119896 119891119896isinL1 (120575
119896) isin Δ such that [119891
119899119896120575119896] = 120588119899 [119891119896120575119896] = 120588
and 119891119899119896rarr 119891119896as 119899 rarr infin 119896 isinN
Applying the Hartley-Hilbert transform for both sidesimplies BA
119891119899119896rarr BA
119891119896in the space of continuous
functions Therefore considering limits we get
B
A[119891119899119896
120575119896
] 997888rarrB
A[119891119896
120575119896
] (56)
This completes the proof of the theorem
Theorem15 TheHartley-Hilbert transformBA is continuouswith respect to the Δ-convergence
Proof Let 120588119899
Δ
997888rarr 120588 as 119899 rarr infin in 120588L1 then there is 119891119899isin L1
and 120575119899isin Δ such that
(120588119899minus 120588) lowast 120575
119899= [119891119899lowast 120575119899
120575119896
] 119891119899997888rarr 0 as 119899 997888rarr infin
(57)
Thus by the aid of Theorem 3 and the hypothesis of thetheorem we have
B
A((120588119899minus 120588) lowast 120575
119899) =B
A[119891119899lowast 120575119899
120575119896
]
997888rarrBA(119891119899lowast 120575119899) as 119899 997888rarr infin
997888rarrBA119891119899
as 119899 997888rarr infin
997888rarr 0 as 119899 997888rarr infin(58)
ThereforeBA(120588119899minus 120588) rarr 0 as 119899 rarr infin ThusBA
120588119899
Δ
997888rarr
BA120588 as 119899 rarr infinThis completes the proof
Lemma 16 Let [119891119899120575119899] isin 120588L1 and 120575 has the usual meaning
of (34) then
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899
120575119899
] (59)
Proof Let 120588 = [119891119899120575119899] isin 120588L1 then
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899lowast 120575
120575119899
]
= lim119899rarrinfin
BA(119891119899lowast 120575)
= lim119899rarrinfin
BA119891119899
(60)
Hence
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899
120575119899
] (61)
Theorem 17 The Hartley-Hilbert transform BA is one-to-one
Proof LetBA[119891119899120575119899] =BA[119892119899120574119899] then by the aid of (46)
we get
lim119899rarrinfin
BA119891119899= lim119899rarrinfin
BA119892119899 (62)
Hence
BA( lim119899rarrinfin
119891119899) =B
A( lim119899rarrinfin
119892119899) (63)
that isBA119891 =BA
119892 The fact thatBA is one-to-one implies119891 = 119892 Hence we have the following theorem
6 Abstract and Applied Analysis
4 A Comparative Study Fourier-HilbertTransform
In [6 7] the Hilbert transform via the Fourier transform of119891(119909) is defined as
BF(119891) (119910)
=1
120587int
infin
0
(F119894(119891) (119909) cos (119909119910) minusF
119903(119891) (119909) sin (119909119910)) 119889119909
(64)
where
F119903(119891) (119909) = int
infin
0
119891 (119905) cos (119909119905) 119889119905
F119894(119891) (119909) = int
infin
0
119891 (119905) sin (119909119905) 119889119905(65)
are respectively the real and imaginary components of theFourier transform of 119891 which are related by
F (119891) (119909) = F119903(119891) (119909) minus 119894F
119894(119891) (119909) (66)
It is interesting to know that a concrete relationship betweenF119903A119890 and F
119894A119900 is described as F
119903(119909) = A119890(119909) and
F119894(119909) = A119900(119909) [2] Those equations above justify the
following statements of the next theorems
Theorem 18 Let [119891119899120575119899] isin 120588L1 then the sequence of Fourier-
Hilbert transforms of (119891119899) satisfies
BF(119891119899) (119910)
=1
120587int
infin
0
(F119894(119891119899) (119909) cos (119909119910) minusF
119903(119891119899) (119909) sin (119909119910)) 119889119909
(67)
and converges uniformly on each compact subsetK ofR
Thus for [119891119899120575119899] isin 120588L1 the Fourier-Hilbert transform of
[119891119899120575119899] is similarly defined by
B
F[119891119899
120575119899
] = lim119899rarrinfin
BF119891119899 (68)
on compact subsets ofRThe following theorems are stated and their proofs are
justified for similar reasons We prefer to we omit the details
Theorem 19 The generalized Fourier-Hilbert transformBF
is linear
Theorem 20 BF(120588 lowast 120598
119899) =BF120588 =BF(120598 lowast 120588) (120598
119899) isin Δ
Theorem 21 IfBF1205881= 0 then 120588
1= 0
Theorem 22 If 120588119899
Δ
997888rarr 120588 as 119899 rarr infin in 120588L1 thenBF120588119899
Δ
997888rarr
BF120588 as 119899 rarr infin in 120588L1 on compact subsets
Theorem 23 The Fourier-Hilbert transformBF is continu-ous with respect to the 120575-convergence
Theorem 24 The Fourier-Hilbert transformBF is continu-ous with respect to the Δ-convergence
Proofs of the above theorems are similar to that given forthe corresponding ones of Hartley-Hilbert transform
References
[1] A H Zemanian Generalized Integral Transformations DoverPublications New York NY USA 2nd edition 1987
[2] R S Pathak Integral Transforms of Generalized Functions andTheir Applications Gordon and Breach Science PublishersNorth Vancouver Canada 1997
[3] S K Q Al-Omari D Loonker P K Banerji and S L KallaldquoFourier sine (cosine) transform for ultradistributions and theirextensions to tempered and ultraBoehmian spacesrdquo IntegralTransforms and Special Functions vol 19 no 5-6 pp 453ndash4622008
[4] S Al-Omari and A Kılıcman ldquoOn the generalized Hartley-Hilbert and fourier-Hilbert transformsrdquo Advances in DifferenceEquations vol 2012 p 232 2012
[5] R P Millane ldquoAnalytic properties of the Hartley transform andtheir applicationsrdquo Proceedings of the IEEE vol 82 no 3 pp413ndash428 1994
[6] N Sundararajan and Y Srinivas ldquoFourier-Hilbert versusHartley-Hilbert transforms with some geophysical applica-tionsrdquo Journal of Applied Geophysics vol 71 no 4 pp 157ndash1612010
[7] N Sundarajan ldquoFourier and hartley transforms amathematicaltwinrdquo Indian Journal of Pure and Applied Mathematics vol 8no 10 pp 1361ndash1365 1997
[8] T K Boehme ldquoThe support of Mikusinski operatorsrdquo Transac-tions of theAmericanMathematical Society vol 176 pp 319ndash3341973
[9] P Mikusinski ldquoFourier transform for integrable BoehmiansrdquoThe Rocky Mountain Journal of Mathematics vol 17 no 3 pp577ndash582 1987
[10] S K Q Al-Omari and A Kılıcman ldquoOn diffraction Fresneltransforms for Boehmiansrdquo Abstract and Applied Analysis vol2011 Article ID 712746 11 pages 2011
[11] P Mikusinski ldquoTempered Boehmians and ultradistributionsrdquoProceedings of the American Mathematical Society vol 123 no3 pp 813ndash817 1995
[12] S K Q Al-Omari and A Kılıcman ldquoNote on Boehmians forclass of optical Fresnel wavelet transformsrdquo Journal of FunctionSpaces and Applications vol 2012 Article ID 405368 14 pages2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Theorem 8 Let 119891 isin L1 then BA(119891 lowast 120575
119899)(119910) rarr BA
119891(119910)
as 119899 rarr infin
Proof For 119891 isinL1 (120575119899) isin Δ then using of (14) implies
BA(119891 lowast 120575
119899) (119910)
= int
infin
minusinfin
(A119900
(119891 lowast 120575119899) (119909) cos119909119910 +A119890 (119891 lowast 120575
119899) (119909) sin119909119910) 119889119909
(37)
Since
(119891 lowast 120575119899) (120577) = int
infin
minusinfin
119891 (119905) 120575119899(120577 minus 119905) 119889119905 997888rarr 119891 (120577) (38)
as 119899 rarr infin we see that
A119900
(119891 lowast 120575119899) (119909) = int
infin
minusinfin
(119891 lowast 120575119899) (120577) sin119909120577119889120577
= int
infin
minusinfin
119891 (119905) int
infin
minusinfin
120575119899(120577 minus 119905) sin (119909120577) 119889120577119889119905
997888rarr int
infin
minusinfin
119891 (119905) sin (119909119905) 119889119905
= A119900
(119891) (119909)
(39)
Similarly
A119890
(119891 lowast 120575119899) (119909) 997888rarr A
119890
(119891) (119909) as 119899 997888rarr infin (40)
Therefore invoking the above equations in (37) we getBA(119891 lowast 120575
119899)(119910) rarr BA
119891(119910) as 119899 rarr infin Hence thetheorem
Denote by 120588L1 the space of integrable Boehmians then120588L1 is a convolution algebra when multiplication by scalaraddition and convolution is defined as [9]
119896 [119891119899
120575119899
] = [119896119891119899
120575119899
]
[119891119899
120575119899
] + [119892119899
120574119899
] = [119891119899lowast 120574119899+ 119892119899lowast 120575119899
120575119899lowast 120574119899
]
[119891119899
120575119899
] lowast [119892119899
120574119899
] = [119891119899lowast 119892119899
120575119899lowast 120574119899
]
(41)
Each function 119891 isin L1 is identified with the Boehmian [119891 lowast120575119899120575119899] Also [119891
119899120575119899] lowast 120575119899= 119891119899isin L1 for every 119899 isin N
Since [120575119899120575119899] corresponds to Dirac delta distribution 120575 the
119896th-derivative of each 120588 isin 120588L1 is defined as
D119896
120588 = 120588 lowastD119896
120575 (42)
The integral of a Boehmian 120588 = [119891119899120575119899] isin 120588L1 is defined as
[11]
int
infin
minusinfin
120588 (119909) 119889119909 = int
infin
minusinfin
1198911(119909) 119889119909 (43)
It is of great interest to observe the following example
Example 9 Every infinitely smooth function119891(119909) isinL1 suchthat D119896119891(119909) notin L1 is integrable as Boehmian but not inte-grable as function
The following has importance in the sense of analysis
Theorem 10 Let [119891119899120575119899] isin 120588L1 then the sequence
BA(119891119899) (119910)
= int
infin
minusinfin
(A119900
119891119899(119909) cos (119909119910) +A119890119891
119899(119909) sin (119909119910)) 119889119909
(44)
converges uniformly on each compact subsetK ofR
Proof By aid of theTheorem 8 and the concept of quotient ofsequences we have
BA(119891119899) (119910) =B
A(119891119899lowast120575119896
120575119896
) (119910)
=BA(119891119899lowast 120575119896
120575119896
) (119910)
=BA(119891119896
120575119896
lowast 120575119899) (119910)
997888rarrBA119891119896
120575119896
(119910) as 119899 997888rarr infin
(45)
where convergence ranges over compact subsets of R Thetheorem is completely proved
Let [119891119899120575119899] isin 120588L1 then by virtue ofTheorem 10 we define
the Hartley-Hilbert transform of the Lebesgue Boehmian[119891119899120575119899] as
B
A[119891119899
120575119899
] = lim119899rarrinfin
BA119891119899 (46)
on compact subsets ofRThe next objective is to establish that our definition is well
defined Let [119891119899120575119899] = [119892
119899120574119899] in 120588L1 then
119891119899lowast 120574119898= 119892119898lowast 120575119899 for every 119898 119899 isinN (47)
Hence applying the Hartley-Hilbert transform to both sidesof the above equation and using the concept of quotients ofsequences imply
BA(119891119899lowast 120574119898) =B
A(119892119898lowast 120575119899) =B
A(119892119899lowast 120575119898) (48)
Thus in particular for 119899 = 119898 and considering Theorem 3we get
lim119899rarrinfin
BA119891119899= lim119899rarrinfin
BA119892119899 (49)
Hence
B
A[119891119899
120575119899
] =B
A[119892119899
120574119899
] (50)
Definition (46) is therefore well defined
Theorem 11 The generalized transformBA is linear
Abstract and Applied Analysis 5
Proof Let 1205881= [119891119899120575119899] and 120588
2= [119892119899120574119899] be arbitrary in 120588L1
and 120572 isin C then 1205881+1205882= [(119891119899lowast120574119899+119892119899lowast120575119899)120575119899lowast120574119899] Hence
employing (46) yields
B
A(1205881+ 1205882) = lim119899rarrinfin
(BA(119891119899lowast 120574119899) +B
A(119892119899lowast 120575119899))
(51)
ByTheorem 8 we get
B
A(1205881+ 1205882) = lim119899rarrinfin
BA119891119899+ lim119899rarrinfin
BA119892119899 (52)
Hence
B
A(1205881+ 1205882) =B
A1205881+B
A1205882 (53)
Also for each complex number 120572 we have
B
A(1205721205881) =B
A[120572119891119899
120575119899
]
= 120572 lim119899rarrinfin
BA119891119899
= 120572B
A1205881
(54)
Hence we have the following theorem
Theorem 12 Let 120588 isin 120588L1 and (120598119899) isin Δ then
B
A(120588 lowast 120598
119899) =B
A120588 =B
A(120598119899lowast 120588) (55)
Proof Let 120588 = [119891119899120575119899] isin 120588L1 then
BA(120588 lowast 120598
119899) =BA[119891119899lowast
120598119899120575119899] = lim
119899rarrinfinBA(119891119899lowast 120598119899)
HenceBA(120588 lowast 120598
119899) = lim
119899rarrinfinBA119891119899=BA120588
Similarly we proceed forBA120588 =BA(120598119899lowast 120588)
This completes the theorem The following theorem isobvious
Theorem 13 IfBA1205881= 0 then 120588
1= 0
Theorem14 TheHartley-Hilbert transformBA is continuouswith respect to the 120575-convergence
Proof Let 120588119899
120575
997888rarr 120588 in 120588L1 as 119899 rarr infin then we show thatBA120588119899
120575
997888rarrBA120588 as 119899 rarr infin Using ([11Theorem26]) we find
119891119899119896 119891119896isinL1 (120575
119896) isin Δ such that [119891
119899119896120575119896] = 120588119899 [119891119896120575119896] = 120588
and 119891119899119896rarr 119891119896as 119899 rarr infin 119896 isinN
Applying the Hartley-Hilbert transform for both sidesimplies BA
119891119899119896rarr BA
119891119896in the space of continuous
functions Therefore considering limits we get
B
A[119891119899119896
120575119896
] 997888rarrB
A[119891119896
120575119896
] (56)
This completes the proof of the theorem
Theorem15 TheHartley-Hilbert transformBA is continuouswith respect to the Δ-convergence
Proof Let 120588119899
Δ
997888rarr 120588 as 119899 rarr infin in 120588L1 then there is 119891119899isin L1
and 120575119899isin Δ such that
(120588119899minus 120588) lowast 120575
119899= [119891119899lowast 120575119899
120575119896
] 119891119899997888rarr 0 as 119899 997888rarr infin
(57)
Thus by the aid of Theorem 3 and the hypothesis of thetheorem we have
B
A((120588119899minus 120588) lowast 120575
119899) =B
A[119891119899lowast 120575119899
120575119896
]
997888rarrBA(119891119899lowast 120575119899) as 119899 997888rarr infin
997888rarrBA119891119899
as 119899 997888rarr infin
997888rarr 0 as 119899 997888rarr infin(58)
ThereforeBA(120588119899minus 120588) rarr 0 as 119899 rarr infin ThusBA
120588119899
Δ
997888rarr
BA120588 as 119899 rarr infinThis completes the proof
Lemma 16 Let [119891119899120575119899] isin 120588L1 and 120575 has the usual meaning
of (34) then
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899
120575119899
] (59)
Proof Let 120588 = [119891119899120575119899] isin 120588L1 then
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899lowast 120575
120575119899
]
= lim119899rarrinfin
BA(119891119899lowast 120575)
= lim119899rarrinfin
BA119891119899
(60)
Hence
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899
120575119899
] (61)
Theorem 17 The Hartley-Hilbert transform BA is one-to-one
Proof LetBA[119891119899120575119899] =BA[119892119899120574119899] then by the aid of (46)
we get
lim119899rarrinfin
BA119891119899= lim119899rarrinfin
BA119892119899 (62)
Hence
BA( lim119899rarrinfin
119891119899) =B
A( lim119899rarrinfin
119892119899) (63)
that isBA119891 =BA
119892 The fact thatBA is one-to-one implies119891 = 119892 Hence we have the following theorem
6 Abstract and Applied Analysis
4 A Comparative Study Fourier-HilbertTransform
In [6 7] the Hilbert transform via the Fourier transform of119891(119909) is defined as
BF(119891) (119910)
=1
120587int
infin
0
(F119894(119891) (119909) cos (119909119910) minusF
119903(119891) (119909) sin (119909119910)) 119889119909
(64)
where
F119903(119891) (119909) = int
infin
0
119891 (119905) cos (119909119905) 119889119905
F119894(119891) (119909) = int
infin
0
119891 (119905) sin (119909119905) 119889119905(65)
are respectively the real and imaginary components of theFourier transform of 119891 which are related by
F (119891) (119909) = F119903(119891) (119909) minus 119894F
119894(119891) (119909) (66)
It is interesting to know that a concrete relationship betweenF119903A119890 and F
119894A119900 is described as F
119903(119909) = A119890(119909) and
F119894(119909) = A119900(119909) [2] Those equations above justify the
following statements of the next theorems
Theorem 18 Let [119891119899120575119899] isin 120588L1 then the sequence of Fourier-
Hilbert transforms of (119891119899) satisfies
BF(119891119899) (119910)
=1
120587int
infin
0
(F119894(119891119899) (119909) cos (119909119910) minusF
119903(119891119899) (119909) sin (119909119910)) 119889119909
(67)
and converges uniformly on each compact subsetK ofR
Thus for [119891119899120575119899] isin 120588L1 the Fourier-Hilbert transform of
[119891119899120575119899] is similarly defined by
B
F[119891119899
120575119899
] = lim119899rarrinfin
BF119891119899 (68)
on compact subsets ofRThe following theorems are stated and their proofs are
justified for similar reasons We prefer to we omit the details
Theorem 19 The generalized Fourier-Hilbert transformBF
is linear
Theorem 20 BF(120588 lowast 120598
119899) =BF120588 =BF(120598 lowast 120588) (120598
119899) isin Δ
Theorem 21 IfBF1205881= 0 then 120588
1= 0
Theorem 22 If 120588119899
Δ
997888rarr 120588 as 119899 rarr infin in 120588L1 thenBF120588119899
Δ
997888rarr
BF120588 as 119899 rarr infin in 120588L1 on compact subsets
Theorem 23 The Fourier-Hilbert transformBF is continu-ous with respect to the 120575-convergence
Theorem 24 The Fourier-Hilbert transformBF is continu-ous with respect to the Δ-convergence
Proofs of the above theorems are similar to that given forthe corresponding ones of Hartley-Hilbert transform
References
[1] A H Zemanian Generalized Integral Transformations DoverPublications New York NY USA 2nd edition 1987
[2] R S Pathak Integral Transforms of Generalized Functions andTheir Applications Gordon and Breach Science PublishersNorth Vancouver Canada 1997
[3] S K Q Al-Omari D Loonker P K Banerji and S L KallaldquoFourier sine (cosine) transform for ultradistributions and theirextensions to tempered and ultraBoehmian spacesrdquo IntegralTransforms and Special Functions vol 19 no 5-6 pp 453ndash4622008
[4] S Al-Omari and A Kılıcman ldquoOn the generalized Hartley-Hilbert and fourier-Hilbert transformsrdquo Advances in DifferenceEquations vol 2012 p 232 2012
[5] R P Millane ldquoAnalytic properties of the Hartley transform andtheir applicationsrdquo Proceedings of the IEEE vol 82 no 3 pp413ndash428 1994
[6] N Sundararajan and Y Srinivas ldquoFourier-Hilbert versusHartley-Hilbert transforms with some geophysical applica-tionsrdquo Journal of Applied Geophysics vol 71 no 4 pp 157ndash1612010
[7] N Sundarajan ldquoFourier and hartley transforms amathematicaltwinrdquo Indian Journal of Pure and Applied Mathematics vol 8no 10 pp 1361ndash1365 1997
[8] T K Boehme ldquoThe support of Mikusinski operatorsrdquo Transac-tions of theAmericanMathematical Society vol 176 pp 319ndash3341973
[9] P Mikusinski ldquoFourier transform for integrable BoehmiansrdquoThe Rocky Mountain Journal of Mathematics vol 17 no 3 pp577ndash582 1987
[10] S K Q Al-Omari and A Kılıcman ldquoOn diffraction Fresneltransforms for Boehmiansrdquo Abstract and Applied Analysis vol2011 Article ID 712746 11 pages 2011
[11] P Mikusinski ldquoTempered Boehmians and ultradistributionsrdquoProceedings of the American Mathematical Society vol 123 no3 pp 813ndash817 1995
[12] S K Q Al-Omari and A Kılıcman ldquoNote on Boehmians forclass of optical Fresnel wavelet transformsrdquo Journal of FunctionSpaces and Applications vol 2012 Article ID 405368 14 pages2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Proof Let 1205881= [119891119899120575119899] and 120588
2= [119892119899120574119899] be arbitrary in 120588L1
and 120572 isin C then 1205881+1205882= [(119891119899lowast120574119899+119892119899lowast120575119899)120575119899lowast120574119899] Hence
employing (46) yields
B
A(1205881+ 1205882) = lim119899rarrinfin
(BA(119891119899lowast 120574119899) +B
A(119892119899lowast 120575119899))
(51)
ByTheorem 8 we get
B
A(1205881+ 1205882) = lim119899rarrinfin
BA119891119899+ lim119899rarrinfin
BA119892119899 (52)
Hence
B
A(1205881+ 1205882) =B
A1205881+B
A1205882 (53)
Also for each complex number 120572 we have
B
A(1205721205881) =B
A[120572119891119899
120575119899
]
= 120572 lim119899rarrinfin
BA119891119899
= 120572B
A1205881
(54)
Hence we have the following theorem
Theorem 12 Let 120588 isin 120588L1 and (120598119899) isin Δ then
B
A(120588 lowast 120598
119899) =B
A120588 =B
A(120598119899lowast 120588) (55)
Proof Let 120588 = [119891119899120575119899] isin 120588L1 then
BA(120588 lowast 120598
119899) =BA[119891119899lowast
120598119899120575119899] = lim
119899rarrinfinBA(119891119899lowast 120598119899)
HenceBA(120588 lowast 120598
119899) = lim
119899rarrinfinBA119891119899=BA120588
Similarly we proceed forBA120588 =BA(120598119899lowast 120588)
This completes the theorem The following theorem isobvious
Theorem 13 IfBA1205881= 0 then 120588
1= 0
Theorem14 TheHartley-Hilbert transformBA is continuouswith respect to the 120575-convergence
Proof Let 120588119899
120575
997888rarr 120588 in 120588L1 as 119899 rarr infin then we show thatBA120588119899
120575
997888rarrBA120588 as 119899 rarr infin Using ([11Theorem26]) we find
119891119899119896 119891119896isinL1 (120575
119896) isin Δ such that [119891
119899119896120575119896] = 120588119899 [119891119896120575119896] = 120588
and 119891119899119896rarr 119891119896as 119899 rarr infin 119896 isinN
Applying the Hartley-Hilbert transform for both sidesimplies BA
119891119899119896rarr BA
119891119896in the space of continuous
functions Therefore considering limits we get
B
A[119891119899119896
120575119896
] 997888rarrB
A[119891119896
120575119896
] (56)
This completes the proof of the theorem
Theorem15 TheHartley-Hilbert transformBA is continuouswith respect to the Δ-convergence
Proof Let 120588119899
Δ
997888rarr 120588 as 119899 rarr infin in 120588L1 then there is 119891119899isin L1
and 120575119899isin Δ such that
(120588119899minus 120588) lowast 120575
119899= [119891119899lowast 120575119899
120575119896
] 119891119899997888rarr 0 as 119899 997888rarr infin
(57)
Thus by the aid of Theorem 3 and the hypothesis of thetheorem we have
B
A((120588119899minus 120588) lowast 120575
119899) =B
A[119891119899lowast 120575119899
120575119896
]
997888rarrBA(119891119899lowast 120575119899) as 119899 997888rarr infin
997888rarrBA119891119899
as 119899 997888rarr infin
997888rarr 0 as 119899 997888rarr infin(58)
ThereforeBA(120588119899minus 120588) rarr 0 as 119899 rarr infin ThusBA
120588119899
Δ
997888rarr
BA120588 as 119899 rarr infinThis completes the proof
Lemma 16 Let [119891119899120575119899] isin 120588L1 and 120575 has the usual meaning
of (34) then
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899
120575119899
] (59)
Proof Let 120588 = [119891119899120575119899] isin 120588L1 then
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899lowast 120575
120575119899
]
= lim119899rarrinfin
BA(119891119899lowast 120575)
= lim119899rarrinfin
BA119891119899
(60)
Hence
B
A([119891119899
120575119899
] lowast 120575) =B
A[119891119899
120575119899
] (61)
Theorem 17 The Hartley-Hilbert transform BA is one-to-one
Proof LetBA[119891119899120575119899] =BA[119892119899120574119899] then by the aid of (46)
we get
lim119899rarrinfin
BA119891119899= lim119899rarrinfin
BA119892119899 (62)
Hence
BA( lim119899rarrinfin
119891119899) =B
A( lim119899rarrinfin
119892119899) (63)
that isBA119891 =BA
119892 The fact thatBA is one-to-one implies119891 = 119892 Hence we have the following theorem
6 Abstract and Applied Analysis
4 A Comparative Study Fourier-HilbertTransform
In [6 7] the Hilbert transform via the Fourier transform of119891(119909) is defined as
BF(119891) (119910)
=1
120587int
infin
0
(F119894(119891) (119909) cos (119909119910) minusF
119903(119891) (119909) sin (119909119910)) 119889119909
(64)
where
F119903(119891) (119909) = int
infin
0
119891 (119905) cos (119909119905) 119889119905
F119894(119891) (119909) = int
infin
0
119891 (119905) sin (119909119905) 119889119905(65)
are respectively the real and imaginary components of theFourier transform of 119891 which are related by
F (119891) (119909) = F119903(119891) (119909) minus 119894F
119894(119891) (119909) (66)
It is interesting to know that a concrete relationship betweenF119903A119890 and F
119894A119900 is described as F
119903(119909) = A119890(119909) and
F119894(119909) = A119900(119909) [2] Those equations above justify the
following statements of the next theorems
Theorem 18 Let [119891119899120575119899] isin 120588L1 then the sequence of Fourier-
Hilbert transforms of (119891119899) satisfies
BF(119891119899) (119910)
=1
120587int
infin
0
(F119894(119891119899) (119909) cos (119909119910) minusF
119903(119891119899) (119909) sin (119909119910)) 119889119909
(67)
and converges uniformly on each compact subsetK ofR
Thus for [119891119899120575119899] isin 120588L1 the Fourier-Hilbert transform of
[119891119899120575119899] is similarly defined by
B
F[119891119899
120575119899
] = lim119899rarrinfin
BF119891119899 (68)
on compact subsets ofRThe following theorems are stated and their proofs are
justified for similar reasons We prefer to we omit the details
Theorem 19 The generalized Fourier-Hilbert transformBF
is linear
Theorem 20 BF(120588 lowast 120598
119899) =BF120588 =BF(120598 lowast 120588) (120598
119899) isin Δ
Theorem 21 IfBF1205881= 0 then 120588
1= 0
Theorem 22 If 120588119899
Δ
997888rarr 120588 as 119899 rarr infin in 120588L1 thenBF120588119899
Δ
997888rarr
BF120588 as 119899 rarr infin in 120588L1 on compact subsets
Theorem 23 The Fourier-Hilbert transformBF is continu-ous with respect to the 120575-convergence
Theorem 24 The Fourier-Hilbert transformBF is continu-ous with respect to the Δ-convergence
Proofs of the above theorems are similar to that given forthe corresponding ones of Hartley-Hilbert transform
References
[1] A H Zemanian Generalized Integral Transformations DoverPublications New York NY USA 2nd edition 1987
[2] R S Pathak Integral Transforms of Generalized Functions andTheir Applications Gordon and Breach Science PublishersNorth Vancouver Canada 1997
[3] S K Q Al-Omari D Loonker P K Banerji and S L KallaldquoFourier sine (cosine) transform for ultradistributions and theirextensions to tempered and ultraBoehmian spacesrdquo IntegralTransforms and Special Functions vol 19 no 5-6 pp 453ndash4622008
[4] S Al-Omari and A Kılıcman ldquoOn the generalized Hartley-Hilbert and fourier-Hilbert transformsrdquo Advances in DifferenceEquations vol 2012 p 232 2012
[5] R P Millane ldquoAnalytic properties of the Hartley transform andtheir applicationsrdquo Proceedings of the IEEE vol 82 no 3 pp413ndash428 1994
[6] N Sundararajan and Y Srinivas ldquoFourier-Hilbert versusHartley-Hilbert transforms with some geophysical applica-tionsrdquo Journal of Applied Geophysics vol 71 no 4 pp 157ndash1612010
[7] N Sundarajan ldquoFourier and hartley transforms amathematicaltwinrdquo Indian Journal of Pure and Applied Mathematics vol 8no 10 pp 1361ndash1365 1997
[8] T K Boehme ldquoThe support of Mikusinski operatorsrdquo Transac-tions of theAmericanMathematical Society vol 176 pp 319ndash3341973
[9] P Mikusinski ldquoFourier transform for integrable BoehmiansrdquoThe Rocky Mountain Journal of Mathematics vol 17 no 3 pp577ndash582 1987
[10] S K Q Al-Omari and A Kılıcman ldquoOn diffraction Fresneltransforms for Boehmiansrdquo Abstract and Applied Analysis vol2011 Article ID 712746 11 pages 2011
[11] P Mikusinski ldquoTempered Boehmians and ultradistributionsrdquoProceedings of the American Mathematical Society vol 123 no3 pp 813ndash817 1995
[12] S K Q Al-Omari and A Kılıcman ldquoNote on Boehmians forclass of optical Fresnel wavelet transformsrdquo Journal of FunctionSpaces and Applications vol 2012 Article ID 405368 14 pages2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
4 A Comparative Study Fourier-HilbertTransform
In [6 7] the Hilbert transform via the Fourier transform of119891(119909) is defined as
BF(119891) (119910)
=1
120587int
infin
0
(F119894(119891) (119909) cos (119909119910) minusF
119903(119891) (119909) sin (119909119910)) 119889119909
(64)
where
F119903(119891) (119909) = int
infin
0
119891 (119905) cos (119909119905) 119889119905
F119894(119891) (119909) = int
infin
0
119891 (119905) sin (119909119905) 119889119905(65)
are respectively the real and imaginary components of theFourier transform of 119891 which are related by
F (119891) (119909) = F119903(119891) (119909) minus 119894F
119894(119891) (119909) (66)
It is interesting to know that a concrete relationship betweenF119903A119890 and F
119894A119900 is described as F
119903(119909) = A119890(119909) and
F119894(119909) = A119900(119909) [2] Those equations above justify the
following statements of the next theorems
Theorem 18 Let [119891119899120575119899] isin 120588L1 then the sequence of Fourier-
Hilbert transforms of (119891119899) satisfies
BF(119891119899) (119910)
=1
120587int
infin
0
(F119894(119891119899) (119909) cos (119909119910) minusF
119903(119891119899) (119909) sin (119909119910)) 119889119909
(67)
and converges uniformly on each compact subsetK ofR
Thus for [119891119899120575119899] isin 120588L1 the Fourier-Hilbert transform of
[119891119899120575119899] is similarly defined by
B
F[119891119899
120575119899
] = lim119899rarrinfin
BF119891119899 (68)
on compact subsets ofRThe following theorems are stated and their proofs are
justified for similar reasons We prefer to we omit the details
Theorem 19 The generalized Fourier-Hilbert transformBF
is linear
Theorem 20 BF(120588 lowast 120598
119899) =BF120588 =BF(120598 lowast 120588) (120598
119899) isin Δ
Theorem 21 IfBF1205881= 0 then 120588
1= 0
Theorem 22 If 120588119899
Δ
997888rarr 120588 as 119899 rarr infin in 120588L1 thenBF120588119899
Δ
997888rarr
BF120588 as 119899 rarr infin in 120588L1 on compact subsets
Theorem 23 The Fourier-Hilbert transformBF is continu-ous with respect to the 120575-convergence
Theorem 24 The Fourier-Hilbert transformBF is continu-ous with respect to the Δ-convergence
Proofs of the above theorems are similar to that given forthe corresponding ones of Hartley-Hilbert transform
References
[1] A H Zemanian Generalized Integral Transformations DoverPublications New York NY USA 2nd edition 1987
[2] R S Pathak Integral Transforms of Generalized Functions andTheir Applications Gordon and Breach Science PublishersNorth Vancouver Canada 1997
[3] S K Q Al-Omari D Loonker P K Banerji and S L KallaldquoFourier sine (cosine) transform for ultradistributions and theirextensions to tempered and ultraBoehmian spacesrdquo IntegralTransforms and Special Functions vol 19 no 5-6 pp 453ndash4622008
[4] S Al-Omari and A Kılıcman ldquoOn the generalized Hartley-Hilbert and fourier-Hilbert transformsrdquo Advances in DifferenceEquations vol 2012 p 232 2012
[5] R P Millane ldquoAnalytic properties of the Hartley transform andtheir applicationsrdquo Proceedings of the IEEE vol 82 no 3 pp413ndash428 1994
[6] N Sundararajan and Y Srinivas ldquoFourier-Hilbert versusHartley-Hilbert transforms with some geophysical applica-tionsrdquo Journal of Applied Geophysics vol 71 no 4 pp 157ndash1612010
[7] N Sundarajan ldquoFourier and hartley transforms amathematicaltwinrdquo Indian Journal of Pure and Applied Mathematics vol 8no 10 pp 1361ndash1365 1997
[8] T K Boehme ldquoThe support of Mikusinski operatorsrdquo Transac-tions of theAmericanMathematical Society vol 176 pp 319ndash3341973
[9] P Mikusinski ldquoFourier transform for integrable BoehmiansrdquoThe Rocky Mountain Journal of Mathematics vol 17 no 3 pp577ndash582 1987
[10] S K Q Al-Omari and A Kılıcman ldquoOn diffraction Fresneltransforms for Boehmiansrdquo Abstract and Applied Analysis vol2011 Article ID 712746 11 pages 2011
[11] P Mikusinski ldquoTempered Boehmians and ultradistributionsrdquoProceedings of the American Mathematical Society vol 123 no3 pp 813ndash817 1995
[12] S K Q Al-Omari and A Kılıcman ldquoNote on Boehmians forclass of optical Fresnel wavelet transformsrdquo Journal of FunctionSpaces and Applications vol 2012 Article ID 405368 14 pages2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of