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 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.