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Obtaining the Fourier spectrum via Fourier coefficients

Marc Carnovale, Jonathan M. Fraser, Ana E. de Orellana

Abstract

The Fourier spectrum is a family of dimensions that interpolates between the Fourier and Hausdorff dimensions and are defined in terms of certain energies which capture Fourier decay. In this paper we obtain a convenient discrete representation of those energies using the Fourier coefficients. As an example application, we use this representation to establish sharp bounds for the Fourier spectrum of a general measure with bounded support, improving previous estimates of the second-named author

Obtaining the Fourier spectrum via Fourier coefficients

Abstract

The Fourier spectrum is a family of dimensions that interpolates between the Fourier and Hausdorff dimensions and are defined in terms of certain energies which capture Fourier decay. In this paper we obtain a convenient discrete representation of those energies using the Fourier coefficients. As an example application, we use this representation to establish sharp bounds for the Fourier spectrum of a general measure with bounded support, improving previous estimates of the second-named author
Paper Structure (14 sections, 9 theorems, 64 equations)

This paper contains 14 sections, 9 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Energy integrals and the Fourier transform
  5. Fourier spectrum
  6. Fourier coefficients and energy integrals
  7. Main result: obtaining the Fourier spectrum via Fourier coefficients
  8. The Fourier transform and the Laplacian operator
  9. Some preliminary estimates involving $L^p$ norms
  10. Proof of Theorem \ref{['thm:coeffs']}
  11. Applications
  12. A discrete formulation for general bounded measures
  13. General bounds for the Fourier spectrum
  14. Bounds involving $\ell^{p}$ spaces

Key Result

Theorem 3.1

Let $\mu$ be a finite Borel measure on $\mathbb{R}^d$ with support contained in $[\delta,1-\delta]^d$ for some $0<\delta<1/2$. Then for all $\theta\in(0,1]$ and $s>0$, In particular,

Theorems & Definitions (17)

  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Corollary 4.1
  • proof
  • Proposition 4.2
  • ...and 7 more