Frame-Like Fourier Expansions for Finite Borel Measures on R

Abstract

It is known that if a finite Borei measure μ on [0,1) possesses a frame of exponential functions for Ζ/2(μ), then μ is of pure type. In this paper, we prove the existence of a class of finite Borei measures μ on [0,1) that are not of pure type that possess frame-like Fourier expansions for L2(μ). We also show properties and classifications of certain measures possessing this type of Fourier expansion. Additionally, we establish a frame-like Fourier expansion for L2(μ) where μ is a singular Borei probability measure on R. Finally, we show measures μ on [0,1) that possess these frame-like Fourier expansions for L2(μ) have all ƒ ϵ L2 (μ) as L2 (μ) limits of harmonic functions with frame-like coefficients. We also discuss when the inner products of these expansions coincide with model spaces and subspaces of harmonic functions on the disk.

Department(s)

Mathematics and Statistics

Keywords and Phrases

finite borei measures; Fourier expansions on the torus; frames; model subspaces; singular measures

International Standard Serial Number (ISSN)

2189-3764; 2189-3756

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2025 Yokohama Publications, All rights reserved.

Publication Date

01 Jan 2024

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