On a relation of distribution with series in L2 and logarithmic averages in the case of symmetric jump behavior

Distribution theory has an important role in applied mathematics, that generalizes the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. Firstly, in the introduction part of this paper we will give some general notations, definitions and results in distribution theory, as  analytic representation of distribution, distributional jump behavior, distributional symmetric jump behavior, tempered distributions, formulas for the jump of distributions in terms of Fourier series, tempered derivative and integral. Then in final part we will state two results, the first one has to do on relation of analytic functions in the upper half-plane with some logarithmic averages in the case of symmetric jump behavior and the second one is related to decomposition of tempered distribution to series