Degree of approximation of functions in the Hölder metric by \((e,c)\) means
Let \(C_{2 \pi}\) denote the Banach space of all \(2 \pi\)-periodic and continuous functions defined on \([- \pi, \pi]\) and let \(H_\alpha\) \((0 < \alpha \leq 1)\) be a subspace of \(C_{2 \pi}\) such that for \(f \in H_\alpha\), \(|f(x) - f(y) |\leq K |x - y |^\alpha\), where \(K > 0\) is some constant. The space \(H_\alpha\) is a Banach space under the norm generated by the Hölder metric [\textit{S. Prössdorf}, Math. Nachr. 69, 7-14 (1975; Zbl 0319.42002)]. The present reviewer [J. Indian Math. Soc., New. Ser. 53, No. 1-4, 99-114 (1988; Zbl 0705.42002) and J. Math. Anal. Appl. 149, No. 1, 236-248 (1990; Zbl 0753.42002)] studied the degree of approximation of functions in Hölder metric by Euler means and Borel means. In this paper, the authors have obtained two theorems concerning the degree of approximation of functions by the \((e,c)\)-mean of its Fourier series in the Hölder metric. One of the theorems reads: Let \(0 < \alpha \leq 1\) and \(0 \leq \beta < \alpha\). Suppose \(f \in H_\alpha\), then \[ \bigl |e_n (f) - f \bigr |_\beta = O(1) \begin{cases} n^{\beta - \alpha} \log n \quad & (0 < \alpha - \beta \leq {1 \over 2}) \\ n^{- {1 \over 2}} \quad & ({1 \over 2} < \alpha - \beta \leq 1) \end{cases}, \] where \(e_n (f;x)\) denotes the \((e,c)\)-mean of \(s_n (f;x)\), the \(n\)th partial sum of the Fourier series of \(f \in H_\alpha\) at \(x\). The other theorem has a different estimate for \(0 < \alpha - \beta \leq {1 \over 2}\) under a Lebesgue-Gergen type condition on \(f\).
scientific article; zbMATH DE number 903367
scientific article