Some results on the analytic representation including the convolution in the Lp spaces

Abstract

In this paper we consider functions in the Lp spaces including the convolution of functions and validity of associative law of the convolution. Using the generalized Cauchy representation, we obtain some results concerning the analytic representation of convolution of functions and distributions. The boundary values representation has been studied for a long time ago and from different points of view. One of the first results is that if we consider three functions from L1 spaces then their convolution has a Cauchy representation.

We obtain several results regarding their convergence or the convergence of the sequence of their analytic representation. We will prove result concerning the analytic representation of the convolution for two functions that are elements of the 1 L spaces and the third function is element of the Lp spaces, then their convolution has a Cauchy representation. Other results, using Fubini’s Theorem, for sequence of functions from L1 spaces and two functions from L1 spaces, their convolution has a Cauchy representation, also has Cauchy representation the boundary value, illustrated with an example. Also we have stated and proved that if a sequence of functions of L1 spaces, which converges to the function in L1(Pn) and a sequence of analytic functions which converges uniformly to the function on every compact subset of the real line, then the sequence of distributions converges to the distribution in D' sense.