Approximation by Nörlund means of Walsh-Fourier series

Abstract

We study the rate of approximation by Nörlund means for Walsh-Fourier series of a function in L p and, in particular, in Lip(α, p) over the unit interval [0, 1), where α > 0 and 1 ⩽ p ⩽ ∞. In case p = ∞, by L p we mean C W , the collection of the uniformly W-continuous functions over [0, 1). As special cases, we obtain the earlier results by Yano, Jastrebova, and Skvorcov on the rate of approximation by Cesàro means. Our basic observation is that the Nörlund kernel is quasi-positive, under fairly general assumptions. This is a consequence of a Sidon type inequality. At the end, we raise two problems.