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