Németh, József

English

Németh József

University of Szeged

A eta Sei. Math., 48 (1985) , 3 7 5 — 3 7 8 Embedding theorems and strong approximation J. NÉMETH Dedicated to Professor Kciroly Tandori on his 60th birthay 1. Let f(x) be a continuous and 2n-periodic function and let (1) f(x) = Z (ancos nx + K sin nx) be its Fourier series. We denote by sn=S„(x) = s„(f; x) the n-th partial sum of (1), and the usual supremum norm by || • ||. We define a class of functions in connection with the strong approximation: s„(X)-= {/• !! i u w i l i / 1=0 where A = {/.„} is a monotonie sequence of positive numbers and 0 It is well known that the classical de la Vallée Poussin means I 2/1 r„ = t „ ( / ; x ) := — 2 «*(*)> n = 1 , 2 , . . . " k=n+l usually approximate the function / , in the supremum norm, better than the partial sums do. Thus it was reasonable that L. L E I N D L E R and A . M E I R [ 2 ] introduced, in analogy to S„(A), the following class of functions: ^ - { / • • I l i ^ i w i ' l H - } . / 1=0 They proved the following result concerning the relation between 5P(A) and VP(X). T h e o r e m ([2, Theorem 1]). If \ and {A„} is a monotonie sequence of positive numbers satisfying the restriction (2) A „ / A 2 „ ^ K n ~ 1, 2, ... Received August 6, 1984. 376 J . N é m e t h with fixed positive K, then (3) sp(;.) c vp(A) holds. Furthermore it is well known that in many cases the classical Fejér means 1 " <r„ = <*„(/; x) := —¡- 2 sk(x) « + 1 k=o approximate the function better than the partial sums do, but worse than the r„ means. Therefore we introduce a new class of functions in connection with the appro-ximation by <7„(x): W ) : = { / : || 2U°n~f\p\\ < 4 n=0 The aim of the present paper is to investigate some relations among this new class and the previous ones. 2. We shall establish the following results. T h e o r e m 1. If p>~ 1 and {A„} is a monotonie nonincerasing sequence of positive numbers, then (4) S p ( / ) c Fp{l) holds. Our Remarks show that the assumptions p > l and /.„J, in certain sense, are necessary. If we omit one or the other (4) does not hold any more. R e m a r k 1. For every p> 1 there exist nondecreasing sequences {A*} and {A**} such that Sp(A*) c Fp()*) and O) SP(1**) * F„{)**). R e m a r k 2. For any 0 t h e r e exist a nonincreasing sequence {A*} and a nondecreasing sequence {/.**} such that S„(/•*) et F , (A*) and Sp(A*+) ct Fp(X**). The proof of these remarks is very elementary and simple, therefore we only present the suitable sequences {A*} and {A**}, and the function f furthermore we detail the proof of (* ) . E m b e d d i n g theorems a n d s t r o n g a p p r o x i m a t i o n 377 T h e o r e m 2. If and {A„} is a monotonie sequence of positive numbers satisfying (2) then (5) Fp(X)<=yp{X) holds. P r o o f of T h e o r e m 1. Since 1 (6) 2K\°n-f\p=2K n = l n + 1 k±o Z ( s k - f ) «=1 n k = o we have to estimate I t from above. Using an inequality of L E I N D L E R (see inequal-ity (8) of [1]) (7) Z K i Z ^ Y ^ K , 2 i t ) n = l k = 1 n=l v k=n > which holds for any / „>0 , a „ ^ 0 and p ^ l , we have (8) «=1 " ~ 1 y —L kÛ k" K where the last inequality follows from the fact that {A„} is monotonic nonincreasing sequence and that p > 1. Inequalities (6) and (8) clearly imply (4), which proves Theorem 1. The first part of Remark 1 instantly follows taking A*=/i (p_1) /2 (using ine-quality (7)). Now we prove the second part of Remark 1. „p-1 Let ;** = " l o g ' ( n + l ) fiFp()*% Since and / ( x ) = 2 cosnx 2KK-f \>= 2 11 = 1 n" 1 We show that f£Sp(À**) and „ t i log" (n + 1 ) " cos kx £ £2 k=n+l K ^ 21 ».p-i / CO l y oo nP-1 1 ~ 1 i L S i f c ' i ~Kn?i logP(n + 1) ^ = M „=1 log"(/i +1) \k =r+i k2) m±i p 1  n" „±i n log (n +1) that is, f£S„(/**). Let x = 0 and N be large enough. Then we get 2 A „ k ( 0 ) - / ( 0 ) r = 2 l o g P ( n + 1 ) • - Î J 2 n" l = i , = r + i I2) ~ N nP-1 I N I 2 , „ , , logp(n + l ) = K 2 — , „ i ! log" 0 + 1) np „tin 378 J. Németh: Embedding theorems and strong approximation which gives that 7,**). Remark 2 can be obtained taking /.*= . , , . ~ cos nx ^ °° COS tlx and f(x)=2 r ~ ; or > - r = 1 o g " and / ( * ) = 2 " ( log n) 1 + p »•=•:1 N" ¿ A RT1+1/P ( l o g n ) 3 / ' Proo f of T h e o r e m 2. If 0 < / ? S l , then using the trivial inequality (9) \a\"^\b\" + \a+b\", we get (10) 2 4 I W I " = 2 4 n = l n = l 2 {St-f) k=n+1 n = l Using condition (2) we obtain ( i i ) A ^ 2 4 2 ( s k - f ) k=0 + n 2 ii 2 0 * - / ) 4 = 0 n A-2 («*- / ) k = 0 2 a 2 W - f ) k = 0 In Kz 2 4 2 ¡*S 2 4 K - / I " . Estimates (10) and (11) give (5) in the case 0 = 1. ' In the case p > \ we use the inequality \a\"^2"-1(\b\" + \a+b\") instead of (9) and we get (12) 2 4 K - / | p s f 2 4 k - / l f similarly as before. Inequality (12) immediately gives assertion (5). References [1] L . LEINDLER, Generalization of inequality of Hardy and Littlewood, Acta Sci. Math., 31 ( 1 9 7 0 ) 2 7 9 — 2 8 5 . [2] L . LEINDLER and A . MEIR, Embedding theorems and strong approximation, Acta Sci. Math., 4 7 ( 1 9 8 4 ) , 3 7 1 — 3 7 5 . JÓZSEF ATTILA UNIVERSITY BOLYAI INSTITUTE ARADI VÉRTANÚK TERE I 6720 SZEGED, H U N G A R Y 

Embedding theorems and strong approximation

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Embedding theorems and strong approximation

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