On the absolute (C,α) convergence for functions of bounded variation

ARTICLE IN PRESS

Journal of Approximation Theory 123 (2003) 300–304 http://www.elsevier.com/locate/jat

Note

On the absolute ðC; aÞ convergence for functions of bounded variation$ Wang Kunyang and Yu Dan Department of Mathematics, Beijing Normal University, Beijing 100875, China Received 21 December 2002; accepted in revised form 7 May 2003 Communicated by Ranko Bojanic

Let f be a 2p-periodic function integrable on ½p; p: For a4  1; the Cesa`ro means of order a of the Fourier series of f are defined by Z 1 p a sn ð f ÞðxÞ ¼ f ðx þ tÞKna ðtÞ dt; p p where Kna ðtÞ ¼

n 1 X Aa1 Dk ðtÞ; a An k¼0 nk

Aan ¼

Gðn þ a þ 1Þ ; Gðn þ 1ÞGða þ 1Þ

Dk ðtÞ ¼

sinðk þ 12Þt : 2 sin 12t

Define Ran ð f ; xÞ ¼

N X

jsak ð f ÞðxÞ  sak1 ð f ÞðxÞj:

k¼nþ1

The following result was presented in [1]: Theorem HB. Let xA½0; p and f a 2p-periodic function of bounded variation on ½p; p: Then, for a40 and nX2; we have n p 4a X Ran ð f ; xÞp V k ðj Þ; np k¼1 0 x $

Supported by NSFC under the Grant 10071007. Corresponding author. Current address (until June 12, 2003): School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia. E-mail address: [email protected] (W. Kunyang). 

0021-9045/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9045(03)00102-3

ARTICLE IN PRESS W. Kunyang, Y. Dan / Journal of Approximation Theory 123 (2003) 300–304

301

where jx ðtÞ :¼ f ðx þ tÞ þ f ðx  tÞ  f ðx þ 0Þ  f ðx  0Þ and Vab ð f Þ denotes the total variation of f on ½a; b: But this is incorrect when 0oao1: Our result is the following theorem. Theorem. Suppose 0oao1; xA½0; p and f is a 2p-periodic function of bounded variation on ½p; p: Then for nX2 n p 100 X Ran ð f ; xÞp 2 a ka1 V0k ðjx Þ; a n k¼1 and there exists a 2p-periodic function f  of bounded variation on ½p; p and a point xA½0; p such that n X p 1 Ran ð f  ; xÞ4 ka1 V0k ðjx Þ ðnX8Þ: a 20000an k¼1

We need two lemmas. Lemma 1. Let 0oao1: Define a gan ðtÞ ¼ Kna ðtÞ  Kn1 ðtÞ 

cos½ðn þ a2Þt  p2a Aan ð2 sin 2t Þa

0otpp:

Then for pnptpp; nX2 16a : jgan ðtÞjp ðntÞaþ1

ð1Þ

Proof. It follows from formula (5.15) of [2, p. 95] that Kna ðtÞ ¼ Dan ðtÞ þ Ena ðtÞ

ð0otppÞ;

where p 1 sin½ðn þ 1þa 2 Þt  2a ; aþ1 a An ð2 sin 2t Þ ( ) N iðnþ12Þt iðnþ1Þt X e e a 1 Aa2 : Ena ðtÞ :¼ R þ n þ 1 ð2 sin t Þ2 Aan ð2 sin 2t Þ n¼nþ1 nþ1 1  eit

Dan ðtÞ :¼

2

Since Dan ðtÞ  Dan1 ðtÞ ¼

pa cos½ðn þ a2Þt  pa a sin½ðn þ a1 2 2 Þt  2   ; a t Aan ð2 sin 2Þ nAan ð2 sin t Þaþ1 2

ARTICLE IN PRESS W. Kunyang, Y. Dan / Journal of Approximation Theory 123 (2003) 300–304

302

we have pa a sin½ðn þ a1 a 2 Þt  2  gan ðtÞ ¼  þ Ena ðtÞ  En1 ðtÞ: nAan ð2 sin 2t Þaþ1

Hence a a jgan ðtÞjp þ jEna ðtÞ  En1 ðtÞj: nAan ð2 sin 2t Þaþ1

ð2Þ

A straightforward calculation gives ( )  it N  a2 X Anþn Aa2 e 2 nþ1þn a a int En ðtÞ  En1 ðtÞ ¼ R  e 2 sin 2t ð1  eit Þ n¼1 Aan1 Aan 

a 1 : nðn þ 1Þ ð2 sin t Þ2 2

Then we obtain jEna ðtÞ



 a 2að1  aÞ þ 3 n sin 2t ð2 sin 2t Þ2 n2   3a p ptpp : o n 2ð2n sin t Þaþ1

a En1 ðtÞjp

1



ð3Þ

2

Substituting (3) into (2) and noticing

1 a Aan pn

(when 0oao1) we get (1).

&

Lemma 2. Let 0oao1: Define Z t a gan ðtÞ :¼ ½Kna ðyÞ  Kn1 ðyÞ dy: 0

Then

8 < 54t; 0otop; jgan ðtÞjp p : 40a1 na1 ta ; otop: n

Proof. By the definition of Kna ðtÞ we easily obtain a jKna ðtÞ  Kn1 ðtÞjp1 þ

This shows jgan ðtÞjp

Z

We have gan ðtÞ ¼ 

n1 aGðn þ 1Þ X ðn  kÞGða þ kÞ p54: Gða þ n þ 1Þ k¼1 Gðk þ 1Þ

t a jKna ðyÞ  Kn1 ðyÞj dyp54 t ð0otopÞ:

0

Z t

p a ½Kna ðyÞ  Kn1 ðyÞ dy ¼ In ðtÞ 

Z

p t

gan ðyÞ dy;

ð4Þ

ARTICLE IN PRESS W. Kunyang, Y. Dan / Journal of Approximation Theory 123 (2003) 300–304

303

where gan is defined in Lemma 1 and Z p cos½ðn þ a2Þy  p2a 1 In ðtÞ ¼ a dy: An t ð2 sin y2Þa Integrating by parts we derive 1 2 jIn ðtÞjp þ a a ðn þ 2ÞAn ð2 sin 2t Þa

Z

cos y2

p t

ð2 sin y2Þ

! dy o aþ1

2 þ 1a 1 : ðn þ a2ÞAan ð2 sin 2t Þa

ð5Þ

By (4) and (5), applying Lemma 1 we get 40 jgan ðtÞjp aþ1 a an t

p when otop: n

&

Proof of the Theorem. We have  N Z p N Z  1 X 1 X a a    Rn ð f ; xÞ ¼ jx ðtÞ dgj ðtÞ ¼   p p j¼nþ1

p

N X

1 p

0

Z

j¼nþ1

0

0

j¼nþ1

p

jgaj ðtÞj dV0t ðjx Þ ¼:

p

 

gaj ðtÞ djx ðtÞ

1 a ðR þ Ran2 Þ; p n1

where Ran1 ¼

N Z X k¼nþ1

0

p k

jgak ðtÞj dV0t ðjx Þ;

Ran2 ¼

Z N X k¼nþ1

By Lemma 2 we have Z p X Z N nþ1 twð0;pkÞ ðtÞ dV0t ðjx Þp54p Ran1 p54 0

k¼nþ1

0

p nþ1

p p k

jgak ðtÞj dV0t ðjx Þ:

p

dV0t ðjx Þ ¼ 54pV0nþ1 ðjx Þ;

where wða;bÞ denotes the characteristic function of ða; bÞ: Also by Lemma 2 we have Z p Z p Z p! N N nþ1 40 X 1 40 X 1 a t dV0 ðjx Þ ¼ þ dV0t ðjx Þ Rn2 p aþ1 a aþ1 a p p p a k¼nþ1 k k t a k¼nþ1 k k t nþ1 0 1 Z p X Z p X N 40@ nþ1 1 1 ¼ dV0t ðjx Þ þ dV0t ðjx ÞA aþ1 ta aþ1 ta p a k p k 0 k4 t nþ1 k¼nþ1 ! Z p p 40 1 1 p 2 V0nþ1 ðjx Þ þ a dV0t ðjx Þ p ta a n nþ1 ! n Z pk X p 40 1 1 ¼ 2 V0nþ1 ðjx Þ þ a dV0t ðjx Þ p ta a n k¼1 kþ1 p

n p 120 X a1 k k V ðjx Þ: 0 a2 na k¼1

ARTICLE IN PRESS 304

W. Kunyang, Y. Dan / Journal of Approximation Theory 123 (2003) 300–304

Combining the estimates for Ran1 and Ran2 we obtain n p 100 X ka1 V0k ðjx Þ ðnX2Þ: Ran ð f ; xÞp 2 a a n k¼1 On the other hand, the function 8 N

when 0oxo2p; when x ¼ 0

gives the second conclusion of the Theorem at point x ¼ p2:

&

Acknowledgments The authors thank Dr. Dai Feng for the valuable discussion.

References [1] N. Humphreys, R. Bojanic, Rate of convergence for the absolutely ðC; aÞ summable Fourier series of functions of bounded variation, J. Approx. Theory 101 (1999) 212–220. [2] A. Zygmund, Trigomometric Series, Vol. I, Cambridge University Press, Cambridge, 1959.