Fourier Series for Singular Measures in Higher Dimensions

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发表日期：

2024年12月4日

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

摘要

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