Characterising Extended Lipschitz Type Conditions with Moduli of Continuity

Abstract

We use the concept of moduli of continuity to give some generalised characterisation theorems in some generalised Holder type spaces of functions over R, characterising certain Lipschitz type conditions on functions in terms of their Fourier transforms. Our main results are inspired by a theorem of Titchmarsh, combined with the notion of moduli of continuity, for characterising Jacobi-Lipschitz conditions in L-2 (R+, d mu,(t)) using approximation by a generalised translation operator.