Fourier Series for Two-Dimensional Singular-Fibered Measures

TL;DR

This paper investigates conditions under which singular planar measures admit 2D Fourier expansions, providing explicit duality, algorithmic computability, and extending beyond traditional affine IFS measures, with applications to fractal Bedford-McMullen carpets.

Contribution

It offers new criteria for Fourier expansions of singular measures in 2D, including non-orthogonal systems and explicit duality, expanding the class of measures beyond affine IFS.

Findings

Established concrete conditions for Fourier expansions of singular measures.

Developed a detailed conditioning-analysis based on marginal and conditional measures.

Applied results to fractal Bedford-McMullen carpets.

Abstract

In this paper we study 2D Fourier expansions for a general class of planar measures $\mu$, generally singular, but assumed compactly supported in $\mathbb{R}^2$. We focus on the following question: When does $L^2(\mu)$ admit a 2D system of Fourier expansions? We offer concrete conditions allowing an affirmative answer to the question for a large class of Borel probability measures, and we present an explicit Fourier duality for these cases. Our 2D Fourier analysis relies on a detailed conditioning-analysis. For a given $\mu$, it is based on the corresponding systems of 1D measures consisting of a marginal measure and associated family of conditional measures computed from $\mu$ by the Rokhlin Disintegration Theorem. Our identified $L^2(\mu)$-Fourier expansions are special in two ways: For our measures $\mu$, the Fourier expansions are generally non-orthogonal, but nonetheless, they lend themselves to algorithmic computations. Second, we further stress that our class of 2D measures $\mu$ considered here go beyond what exists in the literature. In particular, our measures do not require affine iterated function system (IFS) properties, but we do study grid IFS measures in detail and provide some technical criteria guaranteeing their admission of Fourier expansions. Our analyses make use of estimates for the Hausdorff dimensions of the measure supports. An important class of examples addressed in this paper is fractal Bedford-McMullen carpets.