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Fourier series for singular measures in higher dimensions

Chad Berner, John E. Herr, Palle E. T. Jorgensen, Eric S. Weber

Abstract

For multi-variable finite measure spaces, we present in this paper a new framework for non-orthogonal $L^2$ Fourier expansions. Our results hold for probability measures $μ$ with finite support in $\mathbb{R}^d$ that satisfy a certain disintegration condition that we refer to as ``slice-singular''. In this general framework, we present explicit $L^{2}(μ)$-Fourier expansions, with Fourier exponentials having positive Fourier frequencies in each of the d coordinates. Our Fourier representations apply to every $f \in L^2(μ)$, are based on an extended Kaczmarz algorithm, and use a new recursive $μ$ Rokhlin disintegration representation. In detail, our Fourier series expansion for $f$ is in terms of the multivariate Fourier exponentials $\{e_n\}$, but the associated Fourier coefficients for $f$ are now computed from a Kaczmarz system $\{g_n\}$ in $L^{2}(μ)$ which is dual to the Fourier exponentials. The $\{g_n\}$ system is shown to be a Parseval frame for $L^{2}(μ)$. Explicit computations for our new Fourier expansions entail a detailed analysis of subspaces of the Hardy space on the polydisk, dual to $L^{2}(μ)$, and an associated d-variable Normalized Cauchy Transform. Our results extend earlier work for measures $μ$ in one and two dimensions, i.e., $d=1 (μ$ singular), and $d=2 (μ$ assumed slice-singular). Here our focus is the extension to the cases of measures $μ$ in dimensions $d >2$. Our results are illustrated with the use of explicit iterated function systems (IFSs), including the IFS generated Menger sponge for $d=3$.

Fourier series for singular measures in higher dimensions

Abstract

For multi-variable finite measure spaces, we present in this paper a new framework for non-orthogonal Fourier expansions. Our results hold for probability measures with finite support in that satisfy a certain disintegration condition that we refer to as ``slice-singular''. In this general framework, we present explicit -Fourier expansions, with Fourier exponentials having positive Fourier frequencies in each of the d coordinates. Our Fourier representations apply to every , are based on an extended Kaczmarz algorithm, and use a new recursive Rokhlin disintegration representation. In detail, our Fourier series expansion for is in terms of the multivariate Fourier exponentials , but the associated Fourier coefficients for are now computed from a Kaczmarz system in which is dual to the Fourier exponentials. The system is shown to be a Parseval frame for . Explicit computations for our new Fourier expansions entail a detailed analysis of subspaces of the Hardy space on the polydisk, dual to , and an associated d-variable Normalized Cauchy Transform. Our results extend earlier work for measures in one and two dimensions, i.e., singular), and assumed slice-singular). Here our focus is the extension to the cases of measures in dimensions . Our results are illustrated with the use of explicit iterated function systems (IFSs), including the IFS generated Menger sponge for .
Paper Structure (18 sections, 26 theorems, 126 equations)

This paper contains 18 sections, 26 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Slice-singular measures
  4. Fourier series on the torus
  5. Some useful theorems
  6. Direct integrals
  7. Proof of Theorem A
  8. Operator Kaczmarz algorithm
  9. Main proof of Theorem A
  10. Coefficients
  11. Examples of slice singular measures
  12. A Normalized Cauchy Transform in higher dimensions
  13. One Variable Normalized Cauchy Transform
  14. A Two Variable Normalized Cauchy Transform
  15. Images of Normalized Cauchy Transforms
  16. ...and 3 more sections

Key Result

Theorem A

If $\mu$ is an $x_{d}$ slice singular Borel probability measure on $[0,1)^{d}$, then for all $f\in L^{2}(\mu)$, there is a $d$-indexed sequence $\{c_{n_{1},\dots, n_{d}}\}$ such that where the limits are in norm and taken in order starting with the right most series. Furthermore, for each $n_{1},\dots ,n_{d}$, the mapping of $f\to c_{n_{1},\dots, n_{d}}$ is a continuous linear functional, and the

Theorems & Definitions (49)

  • Theorem A
  • Theorem : Rokhlin Disintegration
  • Definition 2.1
  • Definition 2.2
  • Theorem : Herglotz
  • Lemma : De Branges
  • Definition 3.1
  • Definition 3.2
  • Theorem B: Herr, Jorgensen, and Weber
  • Proposition 3.3
  • ...and 39 more