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On the approximation by the generalized de la Vallée Poussin means of Fourier series in the spaces \(W^ rH^{\alpha}\). (English) Zbl 0572.42002

Let \(W^ rH^{\alpha}=\{f:f^{(r-1)}\in AC,\quad f^{(r)}\in H^{\alpha}\},\) \((r=1,2,3,...,0<\alpha \leq 1\), \(f^{(0)}=f)\), \(W^ 0H^{\alpha}=H^{\alpha},\) and let \[ \| f(\circ)\|_{\alpha,r}=\max_{| x| \leq \pi}| f(x)| +\sup_{0<h\leq \pi}\{\max_{| x| \leq \pi}| \Delta_ h^{r+1}f(x)| h^{-\alpha -r}\}. \] The following results are proved: If \(f\in W^ rH^{\alpha}\) \((0<\alpha \leq 1\), \(r=0,1,2,...)\) and \(0\leq \beta <\alpha\), then \[ 1\circ \quad \| 1/\lambda_ n\sum^{n-1}_{k=n-\lambda_ n}S_ k(f;\circ)- f(\circ)\|_{\beta,r}=O((1+\log (2n-\lambda_ n/\lambda_ n))(n- \lambda_ n)^{\beta -\alpha}), \]
\[ 2\circ \quad \| 1/\lambda_ n\sum^{n-1}_{k=n-\lambda_ n}S_ k(f;\circ)- f(\circ)\|_{\beta,r}=O(\lambda_ n^{\beta -\alpha})\quad for\quad \alpha <1,\quad =O((1+\log \lambda_ n)\lambda_ n^{\beta -1})\quad for\quad \alpha =1, \]
\[ 3\circ \quad \| \{1/\lambda_ n\sum^{n- 1}_{k=n-\lambda_ n}| S_ k(f;\circ)-f(\circ)|^ p\}^{1/p}\|_{\beta,r}=O(\lambda_ n^{-1/p}n^{\beta -\alpha +1/p})\quad if\quad \alpha p<1, \]
\[ =O(\lambda_ n^{\beta - \alpha}(1+\log (n/n-\lambda_ n+1))^{1/p})\quad if\quad \alpha p=1,\quad O(\lambda_ n^{-1/p}(n-\lambda_ n+1)^{\beta -\alpha +1/p})\quad if\quad \alpha p>1, \] where \(p>0\), \(\{\lambda_ n\}\) is a non-decreasing sequence of integers such that \(\lambda_ 1=1\), \(\lambda_{n+1}-\lambda_ n\leq 1\), and S(f;\(\circ)\) denotes the k-th partial sum of the Fourier series of f. The above estimations generalize results of S. Prössdorf and Z. Stypiński. Some generalizations of \(1\circ\) and \(2\circ\) were obtained by L. Leindler [Stud. Sci. Math. Hung. 14, 431-439 (1979; Zbl 0495.41007)]. The very general form of \(1\circ\) and \(2\circ\) was very recently published by L. Leindler, A. Meir and V. Totik [Acta Math. Hungarica 45, 441-443 (1985)].
Reviewer: W.Lenski

MSC:

42A10 Trigonometric approximation
42A24 Summability and absolute summability of Fourier and trigonometric series
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

Citations:

Zbl 0495.41007
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References:

[1] AH CCCP 23 (1959) 737–770
[2] Leindler, Acta Sci. Math. 29 pp 1– (1968)
[3] Leindler, Acta Sci. Math. 38 pp 317– (1976)
[4] Prössdorf, Math. Nachr. 69 pp 7– (1975)
[5] Stypinski, Functiones et Approximatio 8 pp 101– (1979)
[6] 1960
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