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Fourier and Beyond: Invariance Properties of a Family of Integral Transforms

Cameron L. Williams, Bernhard G. Bodmann, Donald J. Kouri

Abstract

The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier transform can be uniquely characterized by an intertwining relation with dilations and by having a Gaussian as an eigenfunction. This broadens the perspective to an entire family of Fourier-like transforms that are uniquely identified by the same dilation property and having Gaussian-like functions as eigenfunctions. We show that these transforms share many properties with the Fourier transform, particularly unitarity, periodicity and eigenvalues. We also establish short-time analogues of these transforms and show a reconstruction property and an orthogonality relation for the short-time transforms.

Fourier and Beyond: Invariance Properties of a Family of Integral Transforms

Abstract

The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier transform can be uniquely characterized by an intertwining relation with dilations and by having a Gaussian as an eigenfunction. This broadens the perspective to an entire family of Fourier-like transforms that are uniquely identified by the same dilation property and having Gaussian-like functions as eigenfunctions. We show that these transforms share many properties with the Fourier transform, particularly unitarity, periodicity and eigenvalues. We also establish short-time analogues of these transforms and show a reconstruction property and an orthogonality relation for the short-time transforms.

Paper Structure

This paper contains 10 sections, 9 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A Family of Integral Transforms
  3. The $\Phi_n$ transform and its domain
  4. Developing the $\Phi_n$ transform
  5. Some eigenfunctions of $\Phi_n$
  6. Properties of $\Phi_n$
  7. Preservation of the $L^2$ norm
  8. The Spectrum of $\Phi_n$
  9. The $\Phi_n$ and Fourier-Bessel transforms
  10. The Short-Time $\Phi_n$ Transform

Key Result

Proposition 1

Suppose $\mathcal{T}$ is a bounded integral operator defined on a dense subspace $\mathfrak{X}$ of $L^2(\mathbb{R})$ such that $\mathfrak{X}$ is invariant under each $\mathcal{D}_{\alpha}$, $\alpha\neq 0$, and $\mathcal{T}\mathcal{D}_{\alpha}g = \mathcal{D}_{\alpha^{-1}}\mathcal{T}g$ for all $g\in\m

Figures (1)

  • Figure 1: Commutative diagrams showing the relationships between $\Phi_n^+$ and $\Phi_n^-$ and the Fourier-Bessel transform.

Theorems & Definitions (22)

  • Proposition 1
  • proof
  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 12 more