Partial best approximations and the absolute Cesaro summability of multiple Fourier series
Дербес ең жақсы жуықтаулар және еселi Фурье қатарының абсолюттi чезаролық қосындылануы
Частные наилучшие приближения и абсолютная чезаровская суммируемость кратных рядов Фурье
Bitimkhan S. Alibieva D.T.
2021E.A.Buketov Karaganda State University Publish House
Bulletin of the Karaganda University. Mathematics Series
2021#103Issue 34 - 12 pp.
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 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; β̄|λ-summability of the Fourier series of the function f are obtained. In the first theorem, a sufficient condition for the absolute |C; β̄|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 (Formula Presented) . 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 (Formula Presented) (in the second theorem), (Formula Presented) (in the third theorem), (Formula Presented) (in the fourth theorem).
absolute summability of the series , Fourier series , Lebesgue space , partial best approximation of a function , trigonometric series
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Karagandy University of the name of Academician E.A. Buketov, Karaganda, Kazakhstan
Karagandy University of the name of Academician E.A. Buketov
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