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Non-Abelian Fourier Analysis on $\mathbfΓ\backslash SE(d)$

Arash Ghaani Farashahi, Gregory S. Chirikjian

Abstract

This paper presents a systematic study for the general theory of non-Abelian Fourier series of integrable functions on the homogeneous space $\mathbfΓ\backslash SE(d)$, where $SE(d)$ is the special Euclidean group in dimension $d$, and $\mathbf{\mathbfΓ}$ is a discrete and co-compact subgroup of $SE(d)$. Suppose that $μ$ is the finite $SE(d)$-invariant measure on the right coset space $\mathbfΓ\backslash SE(d)$, normalized with respect to Weil's formula. The analytic aspects of the proposed method works for any given orthonormal basis of the Hilbert function space $L^2(\mathbfΓ\backslash SE(d),μ)$. The paper is concluded with some convolution and Plancherel formulas.

Non-Abelian Fourier Analysis on $\mathbfΓ\backslash SE(d)$

Abstract

This paper presents a systematic study for the general theory of non-Abelian Fourier series of integrable functions on the homogeneous space , where is the special Euclidean group in dimension , and is a discrete and co-compact subgroup of . Suppose that is the finite -invariant measure on the right coset space , normalized with respect to Weil's formula. The analytic aspects of the proposed method works for any given orthonormal basis of the Hilbert function space . The paper is concluded with some convolution and Plancherel formulas.
Paper Structure (16 sections, 30 theorems, 131 equations)

This paper contains 16 sections, 30 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. General notation
  4. Harmonic analysis on $SE(d)$
  5. Fourier analysis on $SE(d)$
  6. Harmonic analysis on $\mathbf{\Gamma}\backslash SE(d)$
  7. Non-Abelian Fourier Series on $\mathbf{\Gamma}\backslash SE(d)$
  8. Absolutely Convergent Non-Abelian Fourier Series on $\mathbf{\Gamma}\backslash SE(d)$
  9. Non-Abelian Fourier Series of Convolutions on $\mathbf{\Gamma}\backslash SE(d)$
  10. Convolution Functions on $\mathbf{\Gamma}\backslash SE(d)$
  11. Non-Abelian Fourier Series of Convolutions
  12. Absolutely Convergent Fourier Series of Convolutions on $\mathbf{\Gamma}\backslash SE(d)$
  13. Plancherel Formula
  14. Examples and Conclusions
  15. The case $\mathbf{\Gamma}\backslash SE(2)$
  16. ...and 1 more sections

Key Result

Proposition 3.1

Let $f\in L^1(SE(d))$ with $\widetilde{|f|}\in L^2(\mathbf{\Gamma}\backslash SE(d),\mu)$. Then $f\in L^2(SE(d))$ and

Theorems & Definitions (52)

  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • proof
  • Proposition 3.7
  • ...and 42 more