1 Sci. Rep. Fukushirna Univ. No.30 (1980)
On Absolute Convergence of Fourier of Class A̲BV'p'
Masaaki SHIBA
f R e e ? f l e d 1 S e p t e m b e r , 1 9 8 0
Series of Function
Let f(x) be an integrable 2π‑periodic function [0,2π) and its Fourier series be
f(:x;)‑ a./2十Σ ̲,(a.cosM‑十bns inn:,,).
If Σ°°,(jan十b 1) < °°, then we say that the Fourier series of f converges absolutely and we write fi∈A. We befine the modulus of continuity of f by
o ((f ;f )= sup lf( ,,1̲ f( :,:) 1
l' ‑ ' く 3
and the integrated modulus of continuity of f by
°p(cf ;f ) e?j?3(S: Ifて'‑1‑h)‑ f(x) 'dx')f'
for p≧1. It is well known that (?p((i f )d" (」ii (? f ) and ??? o ptδ:,f )= o ((i ;f )
We known two famous theorems about absolute convergence of Fourier series in the followings (c.f. [1], [2], and [10]).
THEOREM A. (0.SZASZ) If
Σ二=,n ‑'' (・)2(1/n ;f )< 00 ,
then f∈A.
COROLLARY B.(S.BERNSTEI N) l f
Σ二,n‑''20 (1/n;f )< 00, then f ∈A.
THEOREM C. (A.ZYGMUND) If f is of bounded ,,ariatzon (f∈BV) and
Σ二 n ‑'''Io (1/n ;f )1''',
then f∈A.
Further, we know the bridge theorem between Theorem A and Theorem C (c.f.M.
and S. Izumi [2]). For this, we need the notion of r‑bounded variation (B.V'「), which is defined as follows.
DEFINITION, f is cal led to be of p̲bounded M riation (fi∈BV'p?, when
2
8 M.Shlba :On Absol ute Convergence of Fourier Series of Function of Class A̲BVlp1
:ヨ M> 0; ( Σ:'1f I )l「)''≦M
f or any di1,i si,on e = :x。< ?1,< ̲ < x.= 2n:, ωhere toe pu t I ,= 1:r :x1̲,,] and f(I ,)= fて:t )‑ f(実1 +,)
THOREM D.(M.AND S.IZUM1) 1< p < 00,1/p十1/(j= l and 1 ≦r< 2p.
If f∈BV'「' and
Σ n ̲ ,n ‑ ''‑ ''' l t? . 2 ̲ ,,. (π / n ;f ) 1'‑ 「 f 2p< 0 0 ,
th,on f∈A.
If r= t anh p → 1 then Theorem D reduces to Theorem C. If r →00 and p →00 such that r/p →0, then Theorem D reduces to Theorem A.
In this note,we show the absolute convergence of Fourier series of a function of
A‑bouded variation of order p (A‑BV('))(c.f.M.Shiba[31‑ [5] and D.Waterman [6]‑ [9])
Let A = 1λ.t he a non‑decreasing sequence of positive numbers such that Σλ'= 00 , and let li t be a sequence of non‑overlapping intervals I = ?1x;.y?⊂[0,2π].DEF INIT ION. T he function zs said to be of A ̲bounded vari,ati on of order p (p≧1) , zf 1Σ1f lyn)‑ f(? )l'/λ.1' c(y,,oerges f or et,cry choice of l Inl. The supremun of these sums ts cal led ifle A ̲l,ari ati,on of order p ot f , denoted by VA'p (f ;[0 .2n:))(or VAlp
(f)).
THEOREM. f ∈A̲BV(p, l ≦p< 2r, 1< r< 00 and s‑'十r‑'= 1
If
Σ °°, λ.‑''2'/ n ‑ '2' l o p? l2̲p1s (π/ n ;f ) l '‑ °''< 0 0 ,
then /l∈A.
PROOF OF THEOREM. From Parseval's equality
(t /π)S:[f‑ )‑ fて? h)]dr = 4Σ°°,p sinnh, If =̲a 十b By 2‑‑ p/r 十(2‑ p/r)= p/r 十1(2‑ p)s十pl/s,
? If‑ kπ/N)‑f一 ̲ π /tv)12dr
= ? : If '1,十kπ/N)‑ f (? 十? ,,/N )Ip/「'' 一 一 Sda
≦I S:[f (a,十kπ/N)‑ f(x‑r k‑ 1π/N)]'‑ '一一 ''/ス''d:, l?:N
≦I S: 「 (?十kπ/N )‑ f li ;十k ‑ i π/N)] /λ一 '''x
1? : lf 1,,十kn,/N)‑ f iてi i‑i 二1 11 /N)] ‑ ''d? 1'f ・λN d
" 1o , .s,.。(π/Nf )l' ‑ ' ''・ A N X IS [f (,;‑1‑Io' /N )‑ f (r 十k :: 1π/N「 /λda ?'''
3
Sci. Rep. Fukushima Univ. No.30 (1980) 9
Therefore
ΣN,1? : [f1?十kn:/N)‑ f(r十k‑:1π/N「 d?1'
≦一 o ‑ ・(π/tv;f )12' °Σ '̲ S [f(x 十kπ/N )‑ fて一 k‑ If,:/N)「 /λ.dx
≦ λ,N1(・ｻ,... .。(π/ tv ;f ) 12「 '1VAp (f ;[0 ,2 π)) 1°
≦ C ? .N 1(? ・ +・ (π /N ;n 12「‑ '
So we have
Σ? 10 sin2(nπ/21V)< C? N'「10 1.̲. ̲, (π/N ;f )12‑'/ N‑''「
and then,
Σ n p 2≦ N +'''λN 「loo,̲p.+.( π/N ;f )12‑''「.
Putting Ip .= Σ:k? , then we get 9b・= n''2(Σkp 2)'f2. So,
,j ; ‑‑ n ' f 2S λn f ' 「 10 12 ̲ .? .p (π / n ; f ‑ ) 1 ' ‑ ' '' 「
and
Σ 'p . = Σ 1,i, ‑ ,f; ̲,)/N
= Σ''‑'lj,(1/n十1/n? )十tpN/N
≦Σ '「‑'(1j,n/n )十,PN /N
≦ ? '‑ 'λn'2「1(,o 2̲。̲ (π / N ; f )1‑ °2「/n '‑ ''十 N ' ''? N'''ta ;1 1.+ .(π / N ;f )1'‑pf 「.
By the hypotheses, from N→°°,Σp < °°.
COROLLARY l. f ∈1na1‑BV(pi, 1< p< 2r, 1< r< °°, s‑'十r‑'= 1 and 0< α< 1. If
Σ ' n '‑ °/' 'a ' ‑ 'Io . + 2 ̲ p:,.( π / n ; f ) 1'‑ p' '「< 0 0 ,
then, f ∈A.
Putting λ,= n (0d" a:d" 1) in Theorem, we have Corollary 1.
REMARKS; If p →l and s →°°, Theorem and Corollary l reduce to the results of D.Waterman [6]. If p →°o, and r →o°such that p/r →0, then Theorem and Coroll ry l reduce to Theorem A. If a = 0 in Cororally 1, we have the result of M
and S. Iz umi [2].
If we put s →00, we get the following from Corollary 1,
4
10 M. Shiba :On Absolute Convergence of Fourier Series of Function of Class A̲BV lp1
COROLLARY 2. f ∈1n 1‑BV pi, 1≦p< 2,
Σ °°,n '‑'10 (71 /n ;f )1'‑ f < 00 ,
then f ∈A.
If
The auther wishes to thank Prof. S.Izumi f or his careful reading of this paper
REFERENCES
[1] N.Bari ;Treaties on Trlgonometric series, II , Pergamon, 1964.
[2] M.and S̲Izumi 0n absolute convergence of Fourier series, Arkiv for Math.
7(12),1967, 177‑184.
[3] M.Shiba; 0n Zygmund's theorem for A̲bounded variation of order p, (to appear in Math. bileten).
[4] ̲ ; The uniform convergence of Fourier series of A̲bounded variation, (to appear in Mathematic Analysis).
[5] ̲ ;0n the Fourier coeffcients of a functions of A‑bounded variation (to appear in Pubric. de l'institute Math.).
[6l D.Waterman; 0n convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44, 1972, 107‑117; errata, imbid, 44, 1972,651.
[7] ̲ ; 0n the summability of Fourier series of functions of A‑bounded variation,
・ Studia Math.55 1976 87̲95.
[8] ̲ ; On A̲bounded variation, Studia Math.57, 1976, 33̲45.
[9] ̲ ; Fourier series of functions of A̲bounded variation, Proc. of the Amer.
Math.Soc. 77, 1979, 119̲123.
[10] A.Zygmund; Trigonometric series I, Cambridge, 1959.
Department of Math̲
Faculty of Education Fukushima University,
