Lipschitz classes and convolution approximation processes

Abstract

For a continuous function f(x) on the reals or on the circle T (continuous and 2π periodic) we say that f(x) belongs to the generalized Lipschitz class, denoted by f ∈ Lip* α, ifwhere and Δhf(x) = f(x + ½h)−f(x−½h). For a convolution approximation process given bywherewe shall investigate equivalence relations between the asymptotic behaviour of (d/dx)rAn(f, x) and f ∈ Lip* α.