The generalised jump of a function and Gibbs phenomenon.

By K. G. SHmVASTAW, (Un. Saugor, India)

Summary. - By studying certain transforms and applyi~,g his theorem on • the gsneralised jump of a function, author proves certain theorems co~erni~g jump of a fu~t ion and Gibbs phcaome~a.

1. I n t r o d u c t i o n . - L e t f(0) be in tegrab le in the sense of LV, BES(~U~, in ( - - ~ , ~:). W i t h o u t loss of gene ra l i t y we can take f(0) to be an odd funct ion .

The FOURIER Series of f(0) is then

co

(1.1) f(O) c,~ 2 b, sin nO. 1

The Conjuga te ser ies associa ted wi th (1.1) is

oo

{1.2) 2; b,, sin nO = B,d0).

Here

fit) = - - f ( - t) = - fit + 2=).

W e have

(1.3) ,~(t) = f t~ -}- 0 - - f (~ - - t) - - D, D -~ D(~).

I f for some ~ > 0

(i.@ t o (

~ ( 0 = ~ I t - - U)~-lf(u) d u exis ts

o

and tends to l, as n --* oo then 2l is k n o w n as the genera l i sed j u m p o[ f(t) at t "--" 0.

332 K. C. SrIRIW~STAVa: The gene;ralised jump o] a f'tlJtciioJb etc.

DEFINITION. - If lira sup sn(ts} > l. t , - . ~ 0

Then the series (1.1) is said to exhibit Gr]3BS phenomenon at t------0.

2. We shall make use of the following:

(~.1)

Is

s,~(t) = 2] b, sin vt 1

and

s.(t) = ~ b, c o s vt. 1

Also the first ari thmetic means of series (1.1) and (1.2) are given by

(2.2)

n - - 1

:t

and

1 ~-.-:t ~.(t) = - z s~(O.

n 1

We ~lso define the following notations

(2.3) ( a . = a ~ . -1, r z = z r, ,1 r : = ; r,,-~.

We shall make use of the following Lemma.

I ~ t ~ A . - I f

~+~(t) = o(tp-" a s t - - 0

and if

I d t - - O(log~)for h~ l

then for any [~ > ¢~ 2l

the (3ESARO mean of order (1 q-8) of the sequence t nB~ t tends to - - .

This lemma extends our Theorem [5] in the same direction in which ~ r S A K S ~ S U ~ D ~ [4] generalizes a well known t h e o r e m of GHOW [1]. We are omitt ing the proof as it is analogous to that of ] ~ I N A K S H I S U N D A R A M .

K. C. SnRIVASTAVA: The generalised jump o] a ]unction, etc. 333

We prove here the following theorems.

THEOREM 1. - If for some a < 1 (2.4) holds then

1 4 ( - 1)~.-~ (a) - ~ ' f f " ) - - , e " n 2X-- 1

for

and

n t . - - l ( 2 ) , - - 1 ) u = O ( 1 ) a s tw-.*O

=s, , . if,,) 2 l as n - - oc. (b) 10g n ""

TEEOR~.M 2. - If (2.4) holds for some a < 2 then

ta) t s / ( o ) - ~ ; ( t . ) 1 ---2~

and

(b) 21 f l '

{ # . ( 0 ) - ~.(t.) 1 --* ~ ] u ( 1 - - cos uldu

for

(t. -- . o).

THEOREM 3. - I f (~.4} holds for some ~ < 3, then

2X~

~ . t o ) - ~.~t.) - - ~1 t 1-I~ - oo~ . id . - - ~I fJU

f o r

THEOREM 4 . - Under the same assumption s~(~) cover the whole real axis as t~--~ 0 or

ei ther the limit points of

o

or both s tatements hold.

for n t , - = = O ( ~ )

334 K. C. SHRIVASTAVA: The genera~ised jump of a function, etc.

3. Necessary and sufficient conditions that for any convergent sequence Ts----T i. e. the t ransform

n

T , , - - Y~ a, , , ~ , T~ 1

has a l imit hare

(3.1) lira a~,~--a~ exists for = 1, 2, 3... fcl.**4- ~

n

(a.~) t

and

s

(3.3) lim E a,, ~ -- p exists. f$ --.-~ o o 1

These are Ko#u~A SOHUR [2] conditions of regulari ty. We then have

13.4) T. --.. (0 - - Z a~)T + ~ a~T~ . 1 1

4. We study the t ransform

A . - - 1 ~T~cosv t . ' log n 1 v

Applying ABED' s t ransformation twice we get :

15 ~$--2

(4.1) ~ r ~ = • r~a% + ' T~_~(o._~ -- 20.) + r~c., 1 l

Taking T~ = 1, we get

T~ ---- v, T~ =

and therefore (4.1) becomes

I --~" C#s *

Regarding the t ransformation (4.1) we have the following theorem.

K. C, SHRIVASTAVA: The generalised jump of a /unction, etc. 335

THEOREM B. - Ir (C, 2)T,~ - - T and if

then

nt. --~(2X _ = l o o n )

since T. has a (C, 2) l imit say T then

Taking

therefore

2TJ T as n(n + 1)

cos yr. (~v - - - -

V

no. = cos nt,----- O( 1

Fur ther

so also

and

• log n)

t

~%-~ cos v t - - A%--x __ A2 / ' s in v~dx 0

t

-_ 52v-~ _ _ I /'h2e,~xdx 0

t

o

< t

0

t

< / I 1 - - e '~ l~dx 0

< P

'C 33 ) K. ('. SnrtlVASa'av.l: TI~c gcl~erali.~cd jump of a ] l t n c t i o . . etc. . . . . . . . . . . . . . . . . . . . . .

t h u s we h a v e

a l so the c o n d i t i o n

I A=v-~ cos vt I < 2v-~ + t ~

~, cos vt ~' - v .... ---- 0{11

i s s a t i s f i e d . T h u s tlle t r a n s f o r m a t i o n i s r e g u l a r .

T h e r e f o r e

s Vl~n v2h 21e°s l 1

--= Ollog ~}.

A g a i n

n n

(4.4} v v-~ .~ cos vt = ~ v-~ - - ,, v-~t l - - cos vtt i 1 1

7/ n , ,, 1

---- Z v-* - - 2 v v-* s in- ~ yr. 1 a

: ~ v - ' i _ _ n2t, , z 1

----- (log n + Oqll).

n COS Vt 1 £ - - 1 . .'.._~.lim ~ g n ~ v

T h u s b y 13.4) w e see t h a t A, ~ T.

COROLLARY. - I f nt , = 0 t l ) t h e n

T,, cos vt, c,2 T log n 1 Y

i m p l i e s

W e se t

Tv cos vt, c,z T log n. 1 Y

1~, T,, Ao'= O, to=O, A°'--F, (

v

/ = ~ 1 - , n ~ 1.

t V

K. C. SnRIVASTAVA: The geY~er(dised jump o] a funct ion, etc. 337

By Summat ion by parts we get :

A . log n - - A.'ln cos nt . q- Y~ A./l~ 1 cos yr. - - cos (v q- 1)t. }. I

In order that logari thmic summabil i ty imply A . summabil i ty we must have

(4.5) lim l~ i cos y r . - - cos(v -b 1)t. t - - v exists for v -- 1, 2, 3...

(4.6)

(4,7)

l~' I cos vt, ~ cos (v ~ 1)t. I - - 0(log n) 1

l im [1. cos nt~ q- ~, { cos yr. - - cos (v Jr 1)t. }] : p n ----~ CD 1

also (3.1) is satisfied for a ~ - - 1 . Now (3.6) is equivalent to

which is t rue if n t . - 0(l). Substi tut ing T~----1, in (3.7), we get :

cos vt~ ., c,~ p log n 1 ?

which is t rue for p - - 1 by (4.4). Thus we see that logari thmic Summabil i ty imply A , summabili ty. Fu r the r for (4.8} to hold good it is necessary that n t . - - 0 ( 1 ) and corol-

lary 1 i s proved. From Theorem C and Theorem B and the above Corollary we get

s.(t.) --. ~ (,log n)

f o r

looo) which proves Theorem 1 (b).

5. Next we study the transform,

B. =-i ~. Tv c o s yr.. n l

Annali di Matematica 4 3

338 K. C. SI-IRIVASWAVA: The generalized lump of a function,, etc.

t h e n

THEOREM C. - If (C, 2) T. - - T and if

As b e f o r e

n t . - - ~ ( 2 1 ( - 1 ) ~ = 0 ( 1 )

B. 2{--i)~-1

n m S

n B . - - ~, T~A2o~ + T],~1(o._1 - - 2~,,) + T~o,, 1

N o w

S i m i l a r l y

a n d

on = cos nt.= 0(1).

11\ On-1--COS ( n t . - - t n ) = 0 ~ ) ,

h 2 cos vt = R#~t(1 - - e i t ) 2,

] AScosvtn I < ts,

¢ I AS cos Yr. I < # t . ' = O(n). 1

F r o m (4.2) a n d t3.3j we g e t :

- 1 - n -~ 2; co s vt~ --* - - • t~ ~ c o s vtn --~ p.

n t ~ 1

{ (2X--1)u

tn ~ cos vtn --~ COS z d x 1

0

= ( - - 1) x-1.

al~ Vg --~ o o

w h e r e n o w c~ --- cos v t . . A g a i n A s cos vt n --~ 0 as n ~ oo

K. C~ SI/aIViSTAVA: The generalised lump of a function, etc. 339

Hence

~1 ~ cos yr. 2 (-- 1) x-1 n l = ~ 2 ~ - - i = ~

which proves Theorem C. Tak ing T~ = vb~ and applying Theorem i we get

1 ~ vB~ cos yr, -* 2 1 . 2 (1 --)x-~

i . e.~

a n d

41 (-- 1) x-~ s.'(t.) -- ~: 2), -- 1 "

Thus Theorem 1 (a) is proved.

6. The t ransforms

v. __ ~ T~ (1 - - cos vt.) ~ T

D ° = 1- ~ T~(1 - c o s ~ t , ) .

THEOREM D. - If ~ , '~ 3) T, .... ~'I' _~--d ..if

nt.--2),7: = 0 ( 1 ) ,

then

and

(a) c~ - - T ~ (1 -- u cos u) du

o

(b) D.- T.

By repea ted ABEL'S t ransformat ion we get n--8

1

T._~(~._~ -- 3o.) + " + '

340 K. C. SHRIVASTAVA: The geawralised .lump o] a ]unction, etc.

W e t ake

~ = ~(1 - - cos vtn)

C l e a r l y

h~ (1 - - cos vt,) - - 0 (n - - o% v - - 1, 2, ...).

c,---O(1) fo r ( v - - n , n - - l , n - - 2 , . . . ) .

F u r t h e r

h e n c e

and

$

h~ (1 - - v cos vt) _ _ i fd , t ( 1 _ _ d~)adx " o

t

I h~v-~( 1 - - cos vt) I < j l 1 - - e ~" IAd~ < 0 +~

0

n

E v a I A%- ' (1 - - cos vt,) I < n ' t , 4 - - 0(1). 1

F i n a l l y as in o t h e r t r a n s f o r m s

v-~(i - - cos vt.) - - t. :~ (vt.)-~(l - - cos v~.) 1 1

_ f (1 - uc°s u)du 0

T h u s T h e o r e m D (a) is c o m p l e t e .

A g a i n

(6.1) ha(1 - - cos vt) - - - - R% ~a

= - - ~ t ( 1 - - ea) a

i t f o l l o w s t h a t

(6.2) [ h*( 1 - - cos vt) I < t*

K. C. SHRIVASTAVA: The generalised jump of a function, etc. 341

and

n 1 . t. E (I - - cos yr.). ~; v ~ I ~ ( t - - cos v t j I =

1

Therefore

2).r: 1/ lira ~ v ~t A S " ( 1 - c ° s v t . ) { ~ 2 X u ( 1 - - cos u)du - - 1 .

0

Thus Theorem D tb) is complete. Setting T~--vbv and applying theorem B, we get

i , e ,

~ bv(1-- c°s vt") "-" 2~l~ f ( 1 - cos u) ¢¢

0

l i m { S.(0) - - s.(t.)} 21 f l - - cos u du U

o

and also

lim 1 ~ vb~(1 - - cos yr.) -.- 2_/

i, e,

l im -1 [ s . ' ( 0 ) - s.'(t.}] .-. _21

(6.3) and (6.4) together complete the proof of the theorem.

COROLLARY. - I f (C, 3) T . - - T a n d i f

95t.

then

(6.5) 0 . - - D . = l _ ~ v_~( n _ v)T~(1 - - cos yr.)

T]

342 K. C. SHRIVASTAVA: The generalised lump of a function, etc.

7. Now we shall show that the transform (C, 3) can be replaced by (C, 4). Consider the transform

| - :Z T~(v -'n - - 1)(1 - - cos yr . ) . ~ x

THEOREM E. - If (C, 4) T. -- T and if

then the result (6.5) holds. Set

We have therefore

n t . - 2 ~ = 0 ( 1 ) .

c~ -- (v-~n -- 1)(1 -- cos yr.).

By repeated ABI~,L~ S t ransformation we have

Now

hence

r~ B - - 4

1 1

T • - I " Ca-I" T ; . _ 2 ( c . _ , - - 4~._,) -t- ' + '

1 - - cos yr. = 0 [ 1 )

o

Fur the r from (6,1) and (6.2) we have

I h ' ~ I < n [ h ' v - ' ( l - cos vt.)l + I a ' ( l - cos vt.}l

< nt. 8 -1- t2

hence

r}-

v" 1 a% I < n't°'( n2t. + n) 1

= o ( n ) .

K. C. Snmv.~sT.~v.~: The ge~erati.~ed )ump of a function, etc. 343

H e n c e the condi t ions of r egu l a r i t y are Sat isf ied. Now se t t ing T ~ - vb~, and app ly ing (6.5) we get

1 ~ v-~(n - v)vb~(1 - - cos yr.)

_ [ ~ (~ _ ~)~ + 1 ~ ta - - ~)b~ cos ~tn

= -~.(o) - ~ . ( t~) .

lira [ ~ . ( 0 ) - ~.(t.)] - - ~ t cos .----® u(1 - - u ) - 1

o

T h e r e f o r e

wh ich comple te s the proof of T h e o r e m 3.

1 T~(k sin t . - - sin kvt.).

8. The t r ans fo rm

Let k be a pos i t ive integer , we have iden t ica l ly

k

(8.1} k sin ~¢ - - sin kx -~ sin ~ E { l - - cos (k - - 2v + 1)x }. 1

T a k i n g

k~c,~ = k sin vtn - - s in kvt.

w h e r e n t . = k l : + ¢,, and ), an integer , then u s ing (8.l) we have

k n . c, ----- 2(-- 11~ sin ~. E sin 2 (k - - 2v + llE. + ~ (k "4- 1)),~ . IL

A s s u m e that (K-{-1) ~, is an even number , ----21, > O, then

k . no. - - 2 ( - - 1) ~- s in ~. Y. s in 2 ( k - - 2v q- 1)%

hence a s s u m i n g that

344 K. C. SIIRIVASTh.VA: The generaHsed jump o/ a funvlion, ctc.

w e have :

F u r t h e r

and

kn ] v. [ < kSe.~--O (1-s).

( n - - ~ ) t . = ) , ~ : @ e , - - ~ t , , ~ = 1 , 2,...

k(n - - @)c,-~ - - ( - - 1) ~ { k sin (¢. - - ~t.} - - sin k(t, - - bt. }

/¢

- - 2 ( - - I) )" s in (t. - - ~t.) ~ s in ' • 1

!},21 (k - 2v -[- 1). (~. - - bt.)}t

o(1) o(1) for any f ixed k and ~. Also

W h e r e

H e n c e

h%~ = h ~ sin yr. t A" sin kvt. v k v

and

t

1 A' sin kvt_____. _ _ A ~ / c o s kv~d~ k v

o

t

o

k-~ I ~%-~ s in kvt I < kU ~

1 3.

< 2 k ' t . ' n °

= o(1).

K. C. SHaIVASTAVA: The generolised jump of a ]unvtion,, etc. 345 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L - - . -

Finally

1 ~ sin kvt. - - 1 t. Z sin kvt.

Thus

~ W

I fsin k u -~ ~ J u du.

0

. j ' s i _ p t f s i n k u ~ - - du - - ~ j u du .

1 0 0

(7.1)

for

Applying Theorem h we get the following theorem.

THEORE~ 5, - If (C, 4 t T. ~ T and (k -t- 1)X is an even number then

lira E v - l T ~ { s i n v t . - - s i n k v t . } - - T -~ ! sin u -- s in -~ du 0

nt - - X~: = 0 (~)

as n - - - ~ choosing ), = 1, k odd, and T ~ - vb~ we get

l s.{kt.)] 2l f l ( s i n u s i~ku) {7.2) li.m [s.(t .)--~ -'* r c j u -- . du. 0

Since (C, 4) {nb.)--* 2/ aud k is an odd integer write

lira sup s.(t.) ~-. ~ . t ~ 0

If Y - - - { - - ~ then the limit points of s.(t,} cover the whole real axis, if

Y i s finite then f rom(7.2)as n t , - - 7 : = O ( 1 )

7~

21 [" sin u du ] lira sup l s . ( t . ) - - ~ j T I

0

W

1, 0

Anna l i di Ma tcma t i ca 44

346 K. C. SHRIVASTAVA: The geJ~eralised j,u~p of ~r fU~('t~O~, ~tC"

But k is arbiSrarily large; hence

T~

0

Hence in ei ther case the limit points of s,(t,) contain the intervi~l

"ff 7T

2~lsinud~, ÷21ls inud~ ) ( - - ~ ] u -~ j u 0 0

This and the Theorem A completes the proof of Theorem 4.

9. The results of Theorems 1, 2, 3, 4 also hold under the following ~onditions of generalised jump which have a l ready been proved.

THEOREm. S (a) [6]. - If for a > 0, - - 1 < p < 1, a ~- p > 0

J ~(u) l du = 1 log~

0

as t - -+0

then we have

2l - - - - ----- 0 ( ~ o ) - P

where ~[~+e_l_:(~) stands for i -{- ~ -~ pth (~ESARO mean of the sequence { nB.(x) }.

T m ~ o ~ M S (b) [6]. - I f for o :>0 , - - l < p < l , a - ] - p > 0 , p > - - I and

7T

f I . ' • u ~ t / ! 0

t h e n

21 g ~ + ~ - l ( ~ ) - - ~- "- 0 t ~ o - P ( l o g ¢o) J ' + l ! as ~o - - c ¢ .

Tm~oa~M M [3]. - If

y - - ~ + l O < ~ < y 0 * < a < 2

K. C. Sn~xv.~s'rAvA: The ge~era[ i sed jum~p o] a f u u e t i o n , etc. 347

and

t

:

0

then the sequence nB.(x) is summable to 2/ by C~-i~ mean.

ACKNOWLEDGEMENT

I a m i n d e b t e d to P r o f . M. L. MISRA fo r g u i d a n c e a n d e n c o u r a g e m e n t in the p r e p a r a t i o n of th i s p a p e r .

REFERENCES

[1] CHOW, H. C., On a theorem of O. Szas#, • Journal Load. Math. See. ~, 16 (1941), pp. 29,27. [2] C0OK% R. @., Infinite Matrices and Sequence Spaces, Macmillan Ltd. [3] KINUKAMA MASAKUTI, On the Integro Jump of a Function determined by its Fourier

Coefficients, • Prec. Jap. Academy ,, 3t (1955), 4548. [4] .'~II~AKSHI SUNVERAM S., A note on the Theory of Fourier Series, • Prec. ~at. Inst. Sci.

India ,, 10 (1944), pp. 205.215. [5] SHRIVASTAVA, K. C., On the Determination of the Jump of a Function by iJs Fourier

Coefficients, , ~ohoku Mathematical Journal ,, 12 (1960), pp. 120.129. [6] SULAYANA KUMARI, Determination of the Jump of a Function by its Fourier Series,

•Proc. Nat. Inst. Sci. India ,, 24~ (1958). 204.216 [7] SZASZ OTTO, On the General~sed Jump of a Function and ~ibbs phenomenon, • Duke

Math. J'ourn. ~, 11 (1944), pp. 323.3~3. [8] ZrG~UND, A., Trigonometric Series, Warsaw (1950).