Fourier Series for Two-dimensional Singular-fibered Measures

Abstract

In this paper we study 2D Fourier expansions for a general class of planar measures μ, generally singular, but assumed compactly supported in R2. We focus on the following question: When does L2(μ) 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 μ, it is based on the corresponding systems of 1D measures consisting of a marginal measure and associated family of conditional measures computed from μ by the Rokhlin Disintegration Theorem. Our identified L2(μ)-Fourier expansions are special in two ways: For our measures μ, 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 μ 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.

Department(s)

Mathematics and Statistics

Keywords and Phrases

Fourier series; Fractals; Functional analysis; Iterated function systems; Singular measures

International Standard Serial Number (ISSN)

1096-0813; 0022-247X

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2025 Elsevier, All rights reserved.

Publication Date

15 Jan 2026

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