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Fourier Series for Singular Measures in Higher Dimensions
Journal article   Peer reviewed

Fourier Series for Singular Measures in Higher Dimensions

Chad Berner, John E. Herr, Palle E. T. Jorgensen and Eric S. Weber
The Journal of fourier analysis and applications, Vol.31(1), 1
2025
DOI: 10.1007/s00041-024-10133-8

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Abstract

For multi-variable finite measure spaces, we present in this paper a new framework for non-orthogonal L2 Fourier expansions. Our results hold for probability measures μ with finite support in Rd that satisfy a certain disintegration condition that we refer to as “slice-singular”. In this general framework, we present explicit L2(μ)-Fourier expansions, with Fourier exponentials having positive Fourier frequencies in each of the d coordinates. Our Fourier representations apply to every f∈L2(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 {en}, but the associated Fourier coefficients for f are now computed from a Kaczmarz system {gn} in L2(μ) which is dual to the Fourier exponentials. The {gn} system is shown to be a Parseval frame for L2(μ). Explicit computations for our new Fourier expansions entail a detailed analysis of subspaces of the Hardy space on the polydisk, dual to L2(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.
 Mathematics Partial Differential Equations Abstract Harmonic Analysis Approximations and Expansions Article Fourier Analysis Mathematical Methods in Physics Mathematics and Statistics Signal,Image and Speech Processing

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