Mittag-Leffler Analysis I: Construction and characterization

TL;DR

This paper develops an infinite dimensional analysis framework for non-Gaussian Mittag-Leffler measures, introducing biorthogonal Appell systems and constructing Donsker's delta in this new setting.

Contribution

It introduces a novel non-Gaussian analysis using Mittag-Leffler measures and biorthogonal Appell systems, extending infinite dimensional analysis beyond Gaussian cases.

Findings

Constructed a test function and distribution space for Mittag-Leffler measures.

Developed biorthogonal Appell systems as an alternative to Wick polynomials.

Constructed Donsker's delta as a weak integral in the non-Gaussian setting.

Abstract

We construct an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to this non-Gaussian case. Instead of using Wick ordered polynomials we prove that a system of biorthogonal polynomials, called Appell system, is applicable to the Mittag-Leffler measures. Therefore we are able to introduce a test function and a distribution space. As an application we construct Donsker's delta in a non-Gaussian setting as a weak integral in the distribution space.