Determination of jumps of distributions by differentiated means

Abstract

Differentiated means are defined in order to find formulas for jumps of distributions. We analyze two types of jumps occurring in the notions of distributional jump behavior and symmetric jump behavior. We start by defining what we call Riesz differentiated means for numerical series, then the differentiated means are extended to distributional evaluations for the Schwartz class of tempered distributions. The jumps of tempered distributions are completely determined by the differentiated means of the Fourier transform. We also find formulas for the jumps in terms of the asymptotic behavior of partial derivatives of harmonic representations and harmonic conjugate functions. Applications to Fourier series are given.