Generalized Fourier transforms arising from the enveloping algebras of sl(2) and osp(1|2)

TL;DR

This paper introduces a broad class of generalized Fourier transforms derived from the enveloping algebra of sl(2) and osp(1|2), extending classical Fourier analysis with new operator expressions, kernels, and uncertainty principles.

Contribution

It develops a comprehensive framework for generalized Fourier transforms from algebraic structures, including explicit formulas and kernel decompositions, linking to Clifford-Fourier transforms.

Findings

New class of Fourier transforms satisfying Helmholtz-like properties

Explicit operator exponential formulas for these transforms

Closed-form integral kernels in special cases

Abstract

The Howe dual pair (sl(2),O(m)) allows the characterization of the classical Fourier transform (FT) on the space of rapidly decreasing functions as the exponential of a well-chosen element of sl(2) such that the Helmholtz relations are satisfied. In this paper we first investigate what happens when instead we consider exponentials of elements of the universal enveloping algebra of sl(2). This leads to a complete class of generalized Fourier transforms, that all satisfy properties similar to the classical FT. There is moreover a finite subset of transforms which very closely resemble the FT. We obtain operator exponential expressions for all these transforms by making extensive use of the theory of integer-valued polynomials. We also find a plane wave decomposition of their integral kernel and establish uncertainty principles. In important special cases we even obtain closed formulas for the integral kernels. In the second part of the paper, the same problem is considered for the dual pair (osp(1|2),Spin(m)), in the context of the Dirac operator. This connects our results with the Clifford-Fourier transform studied in previous work.