Leindler, László

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Leindler László

University of Szeged

Acta Sci. Math.. 56 (1992), 287—291 A note on the strong de la Vallée Poussin approximation L. LEINDLER 1. Let {<pn(x)} be an orthonormal system on the finite interval (a, b). We shall consider series (1.1) 2cncpn(x) 11 = 0 with real coefficients satisfying (1.2) ¿ c n 2 < ~ . li=0 By the Riesz—Fischer theorem, series (1.1) converges in the metric Z,2 to a square-integrable function f(x). We denote the n-th partial sum of series (1.1) by sn(x). It is well known that the notion of strong summability is due to H A R D Y and LITTLEWOOD [3], and the notion of strong approximation is due to ALEXITS [2]. Since the strong approximation investigations have started in the 1960s it has become more and more clear that most of the results concerning any property of ordinary approximation have an analogue in strong sense. In other words, we have the same rate of approximation for strong means as for ordinary ones if we consider any one of the most frequently used means. This is true in spite of the facts that, in general, neither strong summability nor strong approximation do not follow from the suitable general ordinary summability and approximation (see M Ó R I C Z [ 1 1 ] and LEINDLER [9]). Some sample theorems showing the great analogy between the ordinary and strong approximation results can be found e.g. in the works [1], [4], [5], [6], [8], [10]. Recently we have discovered that even in the case of the classical de la Vallée Poussin approximation there exists a result which has only a weaker analogue in strong sense. One of the aims of this note is to fill up this gap. In order to formulate our statements precisely we recall some definitions and theorems. Received June 17, 1991. 6» 288 L. Leindler Now we define the ordinary, furthermore the strong and very strong de la Vallée Poussin means with exponent p (p>0): 1 2n-l K(x):=- Z sv(x), (nël), n v=n f 1 2n-l ll/p v„\p-x\:= \ - 2 K(*)-/(*)lp I« v=n ' and {1 2n—1 11/p - 2 M * ) - / ( * ) l p [ , « k - n ' where v:={vt} denotes an arbitrary increasing sequence of positive integers. In [4] we proved Theorem A. Let {A„} be a monotonie sequence of positive numbers such that (1.3) KX\m. *) k=0 If (1.4) B = 0 then we have that K(x)-f(x) = O^n1) holds almost everywhere (a.e.) in (a, b). A similar, but weaker result in connection with the strong approximation (p = 1) was proved in [6] which reads as follows. Theorem B. Let {A„} be a monotonie sequence of positive numbers with (1.3). If (1.5) Z c l X /1=0 then (1.6) V„\p,x\ = ox(A„_1) holds a.e. in (a, b) for any It is easy to see that if A„=nï with y=»0, then conditions (1.4) and (1.5) are equivalent; but if e.g. X„—qn with q > 1, then (1.5) requires much more that (1.4) does in order to have the same order of approximation. *) K will denote positive constant not necessarily the same one at each occurrence. Strong de la Vallée Poussin approximation 289 First we show that (1.4) also always implies (1.6), but this proof will be longer than that of the implication (1.5)=>(1.6) in [6]. Thereafter, using a very general lemma proved only in 1982, we shall extend our result to the very strong approxima-tion ; i.e. we shall prove the following result. Theorem. Let {A„} be a monotonic sequence of positive numbers with (1.3). If (1.4) holds, then (1.7) FJp .v ; x| = ox(K1) a.e. in (a,b) for any 0</>^2 and for any increasing sequence v:={vt} of positive integers. 2. In order to prove our theorem we need two known lemmas. Lemma 1 (Kronecker lemma, see e.g. [1] p. 68). If s„(a) denotes the n-thpartial oo sum of the numerical series am ^ a n increasing sequence of positive num-/71 = 0 oo bers such that A„—°° and the series Z converges, then s„(a)=o(A„) holds. m-0 Lemma 2 ([7]). Let 5 > 0 and {<5„} an arbitrary sequence of non-negative numbers. Suppose that for any orthonormal system {<p„(x)} the condition (2.1) Z U Z c i y — 11=0 m = n implies that the sequence {s„ (x)} of the partial sums of (1.1) possesses a property P, then any subsequence {^(x)} also possesses property P. 3. First we carry out the proof of (1.7) when p=2 and vk=k. An elementary consideration shows that {1 2n—1 1 2n-l 1 - 2 | i v ( * ) - n ( * ) l 2 + - 2 m(*) - / (* ) l 2 • n v=i> n ¥=„ ; The second term on the right hand side of (3.1) is ox(X~2) a.e. in (a, b) on account of Theorem A regarding the monotonicity of the sequence {A„}. Thus we have only to estimate the first term. For that purpose we first show that (3.2) 2 i := 2 ft"'1 f (s„(x)-Va(x)Y n = l a 290 L. Leindler Namely, an easy calculation gives that ' ~ 1 2n—1 21= 2 tin-1-* 2 (2«-v ) 2 c*s n = l » v=n + l ^ 2 ^ cl^ 2 c\ 2 ¿J«-1 ^ 2 № 11 = 1 " v = n + l v = l v/2 <n<v V = 1 whence, by (1.4), (3.2) obviously follows. From (3.2), using B. Levi's theorem, we get that 2 ftn-'isnW-Kix))* n=l a.e. in (a, 6). Hence, by Lemma 1, the estimation 2m (3.3) 2 № ( * ) - KM)2 = 0x(2m) n = l holds a.e. in (a, 6). But (3.3) clearly implies that 1 2m—1 — 2 \sn(x)~K(xr = ox(}->) m n=m also holds a.e. in (a, b). Summing up our partial results we get that (3.4) V„\2>x\ = ox(X-1). On account of the Holder's inequality, we get, for any 0</?^2, that V„\p,x\^V„\2,x\ whence, by (3.4), (3.5) Vn\p,x\ = ox(k~l) also holds a.e. in (a, b). This completes the proof when vk—k. Finally, the statement of Theorem in its generality, i.e. for arbitrary v:={vt}, follows from (3.5) using Lemma 2 with ¿ = 1 and 8„:—X2—(2_1=0); further-more the property P in this case will be just the estimation given by (3.5). Theorem is hereby proved completely. Référencés [1] G. ALEXITS, Konvergenzprobleme der Orthogonalreihen, Akadémiai Kiadó (Budapest, 1960). [2] G. ALEXITS, Sur les bornes de la théorie de l'approximation des fonctions continues par poly-nomes, Magyar Tud. Akad. Mat. Kutató Int. Közi., 8 (1963), 329—340. [3] G. H. HARDY and J. E. LITTLEWOOD, Sur la série de Fourier d'une fonction à carré sommable, Comptes Rendus 28 (1913). Strong de la Vallée Poussin approximation 291 [4] L. LEINDLER, Über die de la Vallée Poussinschen Mittel allgemeiner Orthogonalreihen, Publ. Math. Debrecen, 10 (1963), 274—282. [5] L. LEINDLER, Über die verallgemeinerte de la Vallée Poussinsche Summierbarkeit allgemeiner Orthogonalreihen, Acta Math. Acad. Sei. Hung., 16 (1965), 375—387. [6] L. LEINDLER, On the strong approximation of orthogonal series, Acta Sei. Math., 32 (1971), 41—50. [7] L. LEINDLER, On the extra strong approximation of orthogonal series, Anal. Math., 8 (1982), 125—133. [8] L. LEINDLER, On the generalized strong de la Vallée Poussin approximation, Acta Sei. Math., 56 (1992), 83—88. [9] L. LEINDLER, A note on the relation between ordinary and strong approximation of orthogonal series, Acta Math. Hungarica [10] L. LEINDLER and H. SCHWINN, On the strong and extra strong approximation of orthogonal series, Acta Sei. Math., 45 (1983), 293—304. [11] F. MÓRICZ, A note on the strong 7"-summation of orthogonal series, Acta Sei. Math., 30 (1969), 69—76. BOLYAI INSTITUTE JÓZSEF ATTILA UNIVERSITY ARADI VÉRTANÚK TERE 1 6720 SZEGED, HUNGARY 

A note on the strong de la Vallée Poussin approximation

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A note on the strong de la Vallée Poussin approximation

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