Saturation results for a class of linear operators

Abstract

Let B denote the space of bounded measurable functions with period 2π. We will suppose throughout that f(x) ∈ B. All norms considered are essential sup norms. Let the Fourier series of f(x) be given byLet D = (dnk) (n, k = 0, 1, …) be an infinite matrix.Let Ln(f; x) be the D transform of the Fourier series of f(t) at t = x, i.e.where Sk(x) = A0(x) + A1(x) + … + Ak(x). Let us write