Abstract
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.
Highlights
It is well known that Cesaro means of 2π-periodic functions f ∈ Lp(T) (1 ≤ p ≤ ∞) converges by norms
W1−p (x)dx where p = p/(p − 1) and the supremum is taken over all one-dimensional intervals whose lengths are not greater than 2π
In the present paper we investigate the situation when the weight w can be outside of Ap class
Summary
It is well known that (see [9]) Cesaro means of 2π-periodic functions f ∈ Lp(T) (1 ≤ p ≤ ∞) converges by norms. Hereby T is denoted the interval (−π,π). The problem of the mean summability in weighted Lebesgue spaces has been investigated in [6]. A 2π-periodic nonnegative integrable function w : T → R1 is called a weight function. In the sequel by Lwp (T), we denote the Banach function space of all measurable 2π-periodic functions f , for which. 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). Where p = p/(p − 1) and the supremum is taken over all one-dimensional intervals whose lengths are not greater than 2π.
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