Fourier series for singular measures in higher dimensions

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 ₙ\, but the associated Fourier coefficients for f are now computed from a Kaczmarz system ₙ\ in L²(μ) which is dual to the Fourier exponentials. The ₙ\ 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 $d=3$.