CALDERÓN’S REPRODUCING FORMULAFOR HANKEL CONVOLUTION
R. S. PATHAK AND GIREESH PANDEY
Received 27 March 2006; Accepted 27 March 2006
Calderon-type reproducing formula for Hankel convolution is established using the the-ory of Hankel transform.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Calderon’s formula [3] involving convolutions related to the Fourier transform is usefulin obtaining reconstruction formula for wavelet transform besides many other applica-tions in decomposition of certain function spaces. It is expressed as follows:
f (x)=∫∞
0
(φt ∗φt ∗ f
)(x)
dt
t, (1.1)
where φ :Rn→ C and φt(x)= t−nφ(x/t), t > 0. For conditions of validity of identity (1.1),we may refer to [3].
Hankel convolution introduced by Hirschman Jr. [5] related to the Hankel transformwas studied at length by Cholewinski [1] and Haimo [4]. Its distributional theory wasdeveloped by Marrero and Betancor [6]. Pathak and Pandey [8] used Hankel convolutionin their study of pseudodifferential operators related to the Bessel operator. Pathak andDixit [7] exploited Hankel convolution in their study of Bessel wavelet transforms. Inwhat follows, we give definitions and results related to the Hankel convolution [5] to beused in the sequel.
Let γ be a positive real number. Set
σ(x)= x2γ+1
2γ+1/2Γ(γ+ 3/2),
j(x)= Cγx1/2−γJγ−1/2(x),
Cγ = 2γ−1/2Γ(γ+
12
),
(1.2)
where Jγ−1/2 denotes the Bessel function of order γ− 1/2.
Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2006, Article ID 24217, Pages 1–7DOI 10.1155/IJMMS/2006/24217
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2 Calderon’s formula
We define Lp,σ(0,∞), 1≤ p ≤∞, as the space of those real measurable functions φ on(0,∞) for which
‖φ‖p,σ =[∫∞
0
∣∣φ(x)∣∣pdσ(x)
]1/p
<∞, 1≤ p <∞,
‖φ‖∞,σ = ess sup0<x<∞
|φ(x)| <∞.(1.3)
For each φ∈ L1,σ(0,∞), the Hankel transform of φ is defined by
�(φ)= φ(x)=∫∞
0j(xt)φ(t)dσ(t), 0≤ x <∞. (1.4)
From [5, page 314], we know that φ is bounded and continuous on [0,∞) and
‖φ‖∞,σ ≤ ‖φ‖1,σ . (1.5)
If φ(x)∈ L1,σ(0,∞) and if φ(t)∈ L1,σ(0,∞), then, by inversion, we have [5, page 316]
φ(x)=∫∞
0j(xt)φ(t)dσ(t), 0 < x <∞. (1.6)
If φ(x) and ψ(x) are in L1,σ(0,∞), then the following Parseval formula also holds [10,page 127]:
∫∞0φ(t)ψ(t)dσ(t)=
∫∞0φ(x)ψ(x)dσ(x). (1.7)
To introduce Hankel convolution #, we define
D(x, y,z)=∫∞
0j(xt) j(yt) j(zt)dσ(t)
= 23γ−5/2[Γ(γ+
12
)]2[Γ(γ)π1/2]−1
(xyz)−2γ+1[Δ(xyz)]2γ−2
,
(1.8)
where Δ(xyz) denotes the area of triangle with sides x, y, z if such a triangle exists andzero otherwise. Clearly D(x, y,z)≥ 0 and is symmetric in x, y, z. Applying inversion for-mulae (1.6) to (1.8), we get
∫∞0D(x, y,z) j(zt)dσ(z)= j(xt) j(yt) 0 < x, y <∞, 0≤ t <∞. (1.9)
Now setting t = 0, we obtain∫∞
0D(x, y,z)dσ(z)= 1. (1.10)
Let p,q,r ∈ [1,∞) and 1/r = 1/p+ 1/q− 1. The Hankel convolution of φ ∈ Lp,σ(0,∞) andψ ∈ Lq,σ(0,∞) is defined by [5, page 311]
(φ#ψ)(x)=∫∞
0
∫∞0φ(y)ψ(z)D(x, y,z)dσ(y)dσ(z). (1.11)
R. S. Pathak and G. Pandey 3
By [9, page 179], the integral is convergent for almost all x ∈ (0,∞) and
‖φ#ψ‖r,σ ≤ ‖φ‖p,σ‖ψ‖q,σ . (1.12)
Moreover, if p =∞, then (φ#ψ)(x) is defined for all x ∈ (0,∞) and is continuous. If φ,ψ ∈L1,σ(0,∞), then from [5, page 314]
(φ#ψ)(t)= φ(t)ψ(t), 0≤ t <∞. (1.13)
In this paper, Hankel dilation �a is defined by
�aφ(x)= φa(x)= a−2γ−1φ(x
a
), a > 0. (1.14)
2. Calderon’s formula
In this section, we obtain Calderon’s reproducing identity using the properties of Hankeltransform and Hankel convolutions.
Theorem 2.1. Let φ and ψ ∈ L1,σ(0,∞) be such that following admissibility condition holds:
∫∞0φ(ξ)ψ(ξ)
dσ(ξ)ξ2γ+1 = 1 (2.1)
for all ξ ∈ (0,∞). Then the following Calderon’s reproducing identity holds:
f (x)=∫∞
0
(f #φa#ψa
)(x)
dσ(a)a2γ+1 ∀ f ∈ L1(R). (2.2)
Proof. Taking Hankel transform of the right-hand side of (2.2), we get
�[∫∞
0
(f #φa#ψa
)(x)
dσ(a)a2γ+1
]
=∫∞
0f (ξ)φa(ξ)ψa(ξ)
dσ(a)a2γ+1 = f (ξ)
∫∞0φa(ξ)ψa(ξ)
dσ(a)a2γ+1
= f (ξ)∫∞
0φ(aξ)ψ(aξ)
dσ(a)a2γ+1 = f (ξ).
(2.3)
Now, by putting aξ = ω, we get
∫∞0φ(aξ)ψ(aξ)
dσ(a)a2γ+1 =
∫∞0φ(ω)ψ(ω)
dσ(ω/ξ)(ω/ξ)2γ+1
=∫∞
0φ(ω)ψ(ω)
dσ(ω)ω2γ+1
= 1.
(2.4)
Hence, the result follows.The equality (2.2) can be interpreted in the following L2- sense. �
4 Calderon’s formula
Theorem 2.2. Suppose φ∈ L1,σ(0,∞) is real valued and satisfies
∫∞0
[φ(aξ)
]2 dσ(a)a2γ+1 = 1. (2.5)
For f ∈ L1,σ(0,∞)∩L2,σ(0,∞), suppose that
fε,δ(x)=∫ δε
(f #φa#φa
)(x)
dσ(a)a2γ+1 . (2.6)
Then ‖ f − fε,δ‖2,σ → 0 as ε→ 0 and δ→∞.
Proof. Taking Hankel transform of both sides of (2.6) and using Fubini’s theorem, we get
fε,δ(ξ)= f (ξ)∫ δε
[φ(aξ)
]2 dσ(a)a2γ+1 . (2.7)
By [5, page 311], we have
∥∥φa#φa# f∥∥
2,σ ≤∥∥φa#φa
∥∥1,σ‖ f ‖2,σ
≤ ∥∥φa∥∥21,σ‖ f ‖2,σ .
(2.8)
Now using above inequality and Minkowski’s inequality [2, page 41], we get
‖ f ‖22,σ =
∫∞0dσ(x)
∣∣∣∣∫ δε
(φa#φa# f
)(x)
dσ(a)a2γ+1
∣∣∣∣2
≤∫ δε
∫∞0
∣∣(φa#φa# f)(x)∣∣2dσ(x)
dσ(a)a2γ+1
≤∫ δε
∥∥(φa#φa# f )(x)∥∥
2,σdσ(a)a2γ+1
≤ ∥∥φa∥∥21,σ‖ f ‖2,σ
∫ δε
dt
t
= ∥∥φa‖21,σ‖ f ‖2,σ log
(δ
ε
).
(2.9)
Hence, by Parseval formula (1.7), we get
limε→0δ→∞
‖ f − fε,δ‖22,σ = lim
ε→0δ→∞
‖ f − fε,δ‖22,σ
= limε→0δ→∞
∫∞0
∣∣∣∣ f (ξ)(
1−∫ δε
[φ(aξ)
]2 dσ(a)a2γ+1
)∣∣∣∣2
dσ(x)= 0.(2.10)
Since | f (ξ)(1−∫ δε [φ(aξ)]2(dσ(a)/a2γ+1))|≤| f (ξ)|, therefore, by the dominated conver-gence theorem, the result follows.
The reproducing identity (2.2) holds in the pointwise sense under different sets of niceconditions. �
R. S. Pathak and G. Pandey 5
Theorem 2.3. Suppose f , f ∈ L1,σ(0,∞). Let φ∈ L1,σ(0,∞) be real valued and satisfies
∫∞0
[φ(aξ)
]2 dσ(a)a2γ+1 = 1, ξ ∈R−{0}. (2.11)
Then
limε→0δ→∞
∫ δε
(f #φa#φa
)(x)
dσ(a)a2γ+1 = f (x). (2.12)
Proof. Let
fε,δ(x)=∫ δε
(f #φa#φa
)(x)
dσ(a)a2γ+1 . (2.13)
By [5, page 311], we have
∥∥φa#φa# f∥∥
1,σ ≤∥∥φa#φa
∥∥1,σ
∥∥ f ∥∥1,σ
≤ ∥∥φa∥∥21,σ
∥∥ f ∥∥1,σ .(2.14)
Now,
∥∥ fε,δ∥∥
1,σ =∫∞
0dσ(x)
∣∣∣∣∣∫ δε
(φa#φa# f
)(x)
dσ(a)a2γ+1
∣∣∣∣∣
≤∫ δε
∫∞0
∣∣(φa#φa# f)(x)∣∣dσ(x)
dσ(a)a2γ+1
≤∫ δε
∥∥(φa#φa# f)(x)∥∥
1,σdσ(a)a2γ+1
≤ ∥∥φa∥∥21,σ‖ f ‖1,σ
∫ δε
dt
t
= ∥∥φa∥∥21,σ‖ f ‖1,σ log
(δ
ε
).
(2.15)
Therefore, fε,δ ∈ L1(0,∞). Also using Fubini’s theorem and taking Hankel transform of(2.13), we get
fε,δ(ξ)=∫∞
0j(xξ)
(∫ δε
(φa#φa# f
)(x)
dσ(a)a2γ+1
)dσ(x)
=∫ δε
∫∞0j(xξ)
(φa#φa# f
)(x)dσ(x)
dσ(a)a2γ+1
=∫ δεφa(ξ)φa(ξ) f (ξ)
dσ(a)a2γ+1
= f (ξ)∫ δε
[φ(aξ)
]2 dσ(a)a2γ+1 .
(2.16)
6 Calderon’s formula
Therefore, by (2.11), | fε,δ(ξ)| ≤ | f (ξ)|. It follows that fε,δ ∈ L1,σ(0,∞). By inversion, wehave
f (x)− fε,δ(x)=∫∞
0j(xξ)
[f (ξ)− fε,δ(ξ)
]dσ(ξ), x ∈ (0,∞). (2.17)
Putting
hε,δ(ξ;x)= j(xξ)[f (ξ)− fε,δ(ξ)
]
= j(xξ) f (ξ)[
1−∫ δε
[φ(aξ)
]2 dσ(a)a2γ+1
],
(2.18)
we get
f (x)− fε,δ(x)=∫∞
0j(xξ)
[f (ξ)− fε,δ(ξ)
]dσ(ξ)
=∫∞
0hε,δ(ξ;x)dσ(ξ).
(2.19)
Now using (2.11) in (2.18), we get
limε→0δ→∞
hε,δ(ξ;x)= 0, ξ ∈R−{0}. (2.20)
Since |hε,δ(ξ;x)| ≤ | f (ξ)|, the Lebsegue dominated convergence theorem yields
limε→0δ→∞
[f (x)− fε,δ(x)
]= 0, ∀x. (2.21)
�
Acknowledgment
This work is supported by CSIR (New Delhi), Grant no. 9/13(04)/2003/ EMR-I.
References
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R. S. Pathak and G. Pandey 7
[8] R. S. Pathak and P. K. Pandey, A class of pseudo-differential operators associated with Bessel oper-ators, Journal of Mathematical Analysis and Applications 196 (1995), no. 2, 736–747.
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R. S. Pathak: Department of Mathematics, Banaras Hindu University, Varanasi 221005, IndiaE-mail address: [email protected]
Gireesh Pandey: Department of Mathematics, Banaras Hindu University, Varanasi 221005, IndiaE-mail address: gp2 [email protected]
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