Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Natural geometric Fourier transforms and the associated fractional Laplacian

Razvan M. Tudoran

Abstract

To each arbitrary given general geometric structure on $\mathbb{R}^{n}$, we associate a pair of compatible Fourier transforms, that prove to appear naturally in the framework of Poisson's summation formula for full lattices. We study their properties and the compatibility with the classical $n-$dimensional Fourier transform. In the case of a positive definite geometric structure, we show that these geometric Fourier transforms induce a geometric fractional Laplacian, with properties similar to those of the classical fractional Laplacian.

Natural geometric Fourier transforms and the associated fractional Laplacian

Abstract

To each arbitrary given general geometric structure on , we associate a pair of compatible Fourier transforms, that prove to appear naturally in the framework of Poisson's summation formula for full lattices. We study their properties and the compatibility with the classical dimensional Fourier transform. In the case of a positive definite geometric structure, we show that these geometric Fourier transforms induce a geometric fractional Laplacian, with properties similar to those of the classical fractional Laplacian.
Paper Structure (5 sections, 26 theorems, 77 equations)

This paper contains 5 sections, 26 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Natural geometric structures on $\mathbb{R}^n$ and the associated gradient and Laplace operators
  3. The $n-$dimensional left/right Fourier transforms
  4. The geometric Fourier transforms and Poisson's summation formula for full lattices
  5. A geometric fractional Laplacian

Key Result

Theorem 2.3

TDR Let $b$ be a geometric structure on $\mathbb{R}^n$ and $f\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R})$. Then for any $m\in\mathbb{N}$, the following relation holds true where $\Delta_{b}^{m}:=\Delta_{b}\circ\dots\circ\Delta_{b}$, $m$ times, and $\Delta_{b}^{0}:={\operatorname{Id}}_{\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R})}$.

Theorems & Definitions (34)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Remark 3.5
  • Theorem 3.6
  • Theorem 3.7
  • ...and 24 more