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 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 L^2 () -Fourier expansions, with Fourier exponentials having positive Fourier frequencies in each of the d coordinates. Our Fourier representations apply to every f L² (), are based on an extended Kaczmarz algorithm, and use a new recursive 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 L^2 () which is dual to the Fourier exponentials. The \gₙ\ system is shown to be a Parseval frame for L^2 (). Explicit computations for our new Fourier expansions entail a detailed analysis of subspaces of the Hardy space on the polydisk, dual to L^2 (), and an associated d-variable Normalized Cauchy Transform. Our results extend earlier work for measures in one and two dimensions, i. e. , d=1 (singular), and d=2 (assumed slice-singular). Here our focus is the extension to the cases of measures 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.