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 $μ$ 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 $L^{2}(μ)$-Fourier expansions, with Fourier exponentials having positive Fourier frequencies in each of the d coordinates. Our Fourier representations apply to every $f \in L^2(μ)$, 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 $\{e_n\}$, but the associated Fourier coefficients for $f$ are now computed from a Kaczmarz system $\{g_n\}$ in $L^{2}(μ)$ which is dual to the Fourier exponentials. The $\{g_n\}$ system is shown to be a Parseval frame for $L^{2}(μ)$. Explicit computations for our new Fourier expansions entail a detailed analysis of subspaces of the Hardy space on the polydisk, dual to $L^{2}(μ)$, 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 $d=3$.
The divergence of Mock Fourier series for spectral measures
Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院计算与随机数学教育部重点实验室）
Analytic van der Corput Lemma for p-adic and F_q((t)) oscillatory integrals, singular Fourier transforms, and restriction theorems
本文给出了实vanderCorput引理的p-adic和F_q（（t））模拟，其中相位充分光滑的实条件被相位是收敛的幂级数的条件所取代。这个van der Corput风格的结果，使我们能够在类比的真实情况下，研究奇异的傅里叶变换适当弯曲（解析）流形，并开辟了道路，为进一步的应用。作为这样的进一步应用，我们给出了L^p函数到非阿基米德局部域上适当弯曲的解析流形的傅里叶变换的限制定理，类似于E. Stein和C.弗利曼。
