APPROXIMATION BY NORLUND MEANS OF WALSH-FOURIER SERIES

We study the rate of approximation by Nörlund means for Walsh-Fourier series of a function in Lp and, in particular, in Lip(α, p) over the unit interval [0, 1), where α > 0 and 1 ≤ p ≤ ∞. In case p = ∞, by Lp we mean CW, 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. © 1992.