An Algebraic Approach to Fourier Transformation

TL;DR

This paper develops an algebraic framework for Fourier transformation based on Heisenberg groups, enabling applications in symbolic computation and providing explicit examples like Gaussians and hyperbolic secant functions.

Contribution

It introduces an algebraic approach to Fourier transformation rooted in nilquadratic groups, expanding the theoretical foundation for symbolic computation applications.

Findings

Constructed algebraic structures based on Heisenberg groups.

Analyzed examples including Gaussians and hyperbolic secant algebra.

Established connections with Pontryagin duality.

Abstract

The notion of Fourier transformation is described from an algebraic perspective that lends itself to applications in Symbolic Computation. We build the algebraic structures on the basis of a given Heisenberg group (in the general sense of nilquadratic groups enjoying a splitting property); this includes in particular the whole gamut of Pontryagin duality. The free objects in the corresponding categories are determined, and various examples are given. As a first step towards Symbolic Computation, we study two constructive examples in some detail -- the Gaussians (with and without polynomial factors) and the hyperbolic secant algebra.