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Fuglede's conjecture, differential operators and unitary groups of local translations

Piyali Chakraborty, Dorin Ervin Dutkay, Palle E. T. Jorgensen

Abstract

The purpose of the present paper is to address multiple aspects of the Fuglede question dealing (Fourier spectra vs geometry) with a variety of $L^2$ contexts where we make precise the interplay between the three sides of the question: (i) existence of orthogonal families of Fourier basis functions (and associated spectra) on the one hand, (ii) extensions of partial derivative operators, and (iii) geometry of the corresponding domains, stressing systems of translation-tiles. We emphasize an account of old and new developments since the original 1974-paper by Bent Fuglede where the co-authors and Steen Pedersen have contributed.

Fuglede's conjecture, differential operators and unitary groups of local translations

Abstract

The purpose of the present paper is to address multiple aspects of the Fuglede question dealing (Fourier spectra vs geometry) with a variety of contexts where we make precise the interplay between the three sides of the question: (i) existence of orthogonal families of Fourier basis functions (and associated spectra) on the one hand, (ii) extensions of partial derivative operators, and (iii) geometry of the corresponding domains, stressing systems of translation-tiles. We emphasize an account of old and new developments since the original 1974-paper by Bent Fuglede where the co-authors and Steen Pedersen have contributed.
Paper Structure (12 sections, 22 theorems, 179 equations)

This paper contains 12 sections, 22 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bent Fuglede: Spectral sets and joint spectra
  4. More on the Nikodym restriction
  5. Steen Pedersen: Unbounded spectral sets, generalized eigenvectors and unitary groups
  6. Unions of intervals
  7. Examples
  8. Appendix: Spectral Theorems and some fundamental inequalities
  9. Spectral Theorems, Stone's Theorem
  10. Some fundamental inequalities
  11. Hilbert-Schmidt operators
  12. A stronger form of the Spectral Theorem

Key Result

Theorem 3.1

Fug74 Let $\Omega\subset\mathbb{R}^n$ be a finite measure open and connected Nikodym region (see Definition defnik).

Theorems & Definitions (52)

  • Remark 2.1
  • Definition 2.2
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['th2.1']}
  • Remark 3.3
  • Definition 5.1
  • Definition 5.2
  • Proposition 5.3
  • ...and 42 more