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A novel definition of real Fourier transform

Fulvio Sbisà

TL;DR

The paper introduces a real-valued Fourier transform $\mathscr{F}_{\mathbb{R}}$ that maps real functions to real outputs, avoiding the two-component representation of the standard real/imaginary decomposition. It constructs this transform via a symmetric-antisymmetric decomposition on the Schwartz space, defines $\mathscr{F}_{\mathbb{R}}$ and its inverse through $S$ and $D$, and proves an inversion theorem with $\mathscr{F}_{\mathbb{R}}^{a}$ coinciding with $\mathscr{F}_{\mathbb{R}}$, making it an involution. The framework is shown to be unitary on $L^2$, extendable to $L^2$ and $L^1 \to C_0$, with eigenfunctions given by Hermite functions, and it yields explicit convolution relations that involve the parity operator. The approach preserves parity, provides a natural modal expansion via real-valued kernels, and offers potential simplifications for symmetric problems, while maintaining essential Fourier-analytic properties such as inversion, unitarity, and convolution duality.

Abstract

We propose a novel definition of Fourier transform, with the property that the transform of a real function is again a real function (without doubling the number of real components). We prove the inversion theorem for the novel definition, and show that it shares the good properties of the usual definition.

A novel definition of real Fourier transform

TL;DR

The paper introduces a real-valued Fourier transform that maps real functions to real outputs, avoiding the two-component representation of the standard real/imaginary decomposition. It constructs this transform via a symmetric-antisymmetric decomposition on the Schwartz space, defines and its inverse through and , and proves an inversion theorem with coinciding with , making it an involution. The framework is shown to be unitary on , extendable to and , with eigenfunctions given by Hermite functions, and it yields explicit convolution relations that involve the parity operator. The approach preserves parity, provides a natural modal expansion via real-valued kernels, and offers potential simplifications for symmetric problems, while maintaining essential Fourier-analytic properties such as inversion, unitarity, and convolution duality.

Abstract

We propose a novel definition of Fourier transform, with the property that the transform of a real function is again a real function (without doubling the number of real components). We prove the inversion theorem for the novel definition, and show that it shares the good properties of the usual definition.

Paper Structure

This paper contains 16 sections, 11 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Inversion theorem, involutivity and parity
  3. Def-inition and inversion theorem
  4. Integral representation and involutivity
  5. Fourier transform and parity
  6. Unitarity, extensions and eigenfunctions
  7. Unitarity
  8. Extension theorems
  9. Eigenfunctions
  10. Convolution and Fourier transform
  11. The notion of convolution
  12. Convolution and real Fourier transform
  13. Comparison and spherical symmetry
  14. Conclusions
  15. Proof of the proposition \ref{['prop: real Fourier transform of product and convolution']}
  16. ...and 1 more sections

Key Result

Theorem 1

The linear operators $\mathscr{F}_{_{\!\mathbb{R}}}$ and $\mathscr{F}^{a}_{_{\!\mathbb{R}}}$ are isomorphisms, and where $\textup{Id}$ is the identity operator on $\mathscr{S}$.

Theorems & Definitions (23)

  • Definition 1: real Fourier transform and anti-transform
  • Theorem 1: inversion theorem
  • proof
  • Proposition 1: involutivity of the real Fourier transform
  • proof
  • Proposition 2
  • proof
  • Proposition 3: unitarity of the real Fourier transform
  • proof
  • Corollary 1
  • ...and 13 more