# Orthogonal Fourier Analysis on Domains

**Authors:** Mihail N. Kolountzakis  
**arXiv:** [2408.15361](https://arxiv.org/abs/2408.15361)  
**Date:** 2024-08-27  
**Summary page:** https://www.summarizepaper.com/en/paper/2408.15361v3/

## Key points

- Exploration of the history of the Fuglede or Spectral Set Conjecture over the past 30 years
- Focus on Fourier Analysis on domains with orthogonal bases of exponentials and the Fuglede Conjecture
- Acknowledgment that the paper does not cover developments in classes of finite groups or spectral measures
- Discussion on tiling by translation, including tilings by functions, examples from mathematical literature, lattices, and the Poisson Summation Formula
- Examination of structure results about tilings via the Fourier Transform extending to higher dimensions
- Exploration of spectrality as a tiling question, lattice Fuglede Conjecture proof, and scenarios involving filling a box with bricks of different types
- Discussion on prehistory of the Fuglede Conjecture including non-symmetric convex bodies not being spectral and Turán extremal problem related to positive definite functions with given support
- Section on open questions and unresolved issues related to the Fuglede Conjecture

## Summary

In this paper, Mihail N. Kolountzakis explores the history of the Fuglede or Spectral Set Conjecture over the past 30 years. The focus is on Fourier Analysis on domains with orthogonal bases of exponentials and the Fuglede Conjecture itself. The author acknowledges that the paper is not exhaustive and does not cover developments in classes of finite groups or spectral measures. The selection of material is influenced by personal preference, historical significance, and expertise. The paper delves into tiling by translation, including discussions on tilings by functions, examples from mathematical literature, lattices, and the Poisson Summation Formula. It also examines structure results about tilings via the Fourier Transform and extends to higher dimensions. The concept of spectrality as a tiling question is explored, along with a proof of the lattice Fuglede Conjecture and scenarios involving filling a box with bricks of different types. Additionally, the prehistory of the Fuglede Conjecture is discussed, touching upon topics such as non-symmetric convex bodies not being spectral and the Turán extremal problem related to positive definite functions with given support. The paper concludes with a section on open questions and unresolved issues in relation to the Fuglede Conjecture. Key words associated with this research include Fuglede Conjecture, Spectral Sets, and Tilings. The author pays tribute to Bent Fuglede for his contributions to this area of study. Date: October 31st 2024.