Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: ON THE MEAN SUMMABILITY BY CESARO METHOD OF FOURIER TRIGONOMETRIC SERIES IN TWO-WEIGHTED SETTING | ON THE MEAN SUMMABILITY BY CESARO METHOD OF FOURIER TRIGONOMETRIC SERIES IN TWO-WEIGHTED SETTING A. GUVEN AND V. KOKILASHVILI Received 26 June 2005 Revised 19 October 2005 Accepted 23 October 2005 The Cesaro summability of trigonometric Fourier series is investigated in the weighted Lebesgue spaces in a two-weight case for one and two dimensions. These results are applied to the prove of two-weighted Bernstein s inequalities for trigonometric polynomials of one and two variables. Copyright 2006 A. Guven and V. Kokilashvili. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction It is well known that see 9 Cesaro means of 2n-periodic functions f e Lp T 1 p to converges by norms. Hereby T is denoted the interval -n n . The problem of the mean summability in weighted Lebesgue spaces has been investigated in 6 . A 2n-periodic nonnegative integrable function w T R1 is called a weight function. In the sequel by lW T we denote the Banach function space of all measurable 2n-periodic functions f for which 1 p IIf llp w JT I f x pw x dx to. In the paper 6 it has been done the complete characterization of that weights w for which Cesaro means converges to the initial function by the norm of Lwp T . Later on Muckenhoupt see 3 showed that the condition referred in 6 is equivalent to the condition Ap that is 1 sup m L w1 p x dx to where p p p - 1 and the supremum is taken over all one-dimensional intervals whose lengths are not greater than 2n. Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 41837 Pages 1-15 DOI JIA 2006 41837 2 Mean summability of Fourier trigonometric series The problem of mean summability by linear methods of multiple Fourier trigonometric series in lW T in the frame of Ap classes has been studied in 5 . In the present paper we .
