Partial best approximations and the absolute Cesaro summability of multiple Fourier series

Abstract

The article is devoted to the problem of absolute Cesaro summability of multiple trigonometric Fourier series. Taking a central place in the theory of Fourier series this problem was developed quite widely in the one-dimensional case and the fundamental results of this theory are set forth in the famous monographs by N.K. Bari, A. Zigmund, R. Edwards, B.S. Kashin and A.A. Saakyan [1–4]. In the case of multiple series, the corresponding theory is not so well developed. The multidimensional case has own specifics and the analogy with the one-dimensional case does not always be unambiguous and obvious. In this article, we obtain sufficient conditions for the absolute summability of multiple Fourier series of the function f ∈ Lq(Is) in terms of partial best approximations of this function. Four theorems are proved and four different sufficient conditions for the |C; β¯|λ-summability of the Fourier series of the function f are obtained. In the first theorem, a sufficient condition for the absolute |C; β¯|λ- summability of the Fourier series of the function f is obtained in terms of the partial best approximation of this function which consists of s conditions, in the case when β1 = ... = βs = 1/q'. Other sufficient conditions are obtained for double Fourier series. Sufficient conditions for the |C; β1; β2|λ-summability of the Fourier series of the function f ∈ Lq(I2) are obtained in the cases β1 = 1/q', −1 < β2 < 1/q'(in the second theorem), 1/q'< β1 < +∞, β2 = 1/q', (in the third theorem), −1 < β1 < 1/q', 1/q' < β2 < +∞ (in the fourth theorem).