Tempered Generalized Functions and Hermite Expansions

TL;DR

This paper introduces a new algebra of tempered generalized functions using Hermite expansions, extending Fourier analysis and applying it to stochastic calculus.

Contribution

It develops a novel algebraic framework for tempered distributions via Hermite expansions, enabling natural Fourier transform extension and stochastic analysis applications.

Findings

Extended Fourier transform to the new algebra

Established properties of association, embedding, and point values

Provided a generalized Ito formula for stochastic analysis

Abstract

In this work we introduce a new algebra of tempered generalized functions. The tempered distributions are embedded in this algebra via their Hermite expansions. The Fourier transform is naturally extended to this algebra in such a way that the usual relations involving multiplication, convolution and differentiation are valid. We study the elementary properties of the association, embedding, point values and Fourier transform. Furthermore, we give a generalized Ito formula in this context and some applications to stochastic analysis.