Title

Orthogonal exponentials, translations, and Bohr completions

Authors

Authors

D. E. Dutkay; D. G. Han;P. E. T. Jorgensen

Comments

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Abbreviated Journal Title

J. Funct. Anal.

Keywords

Hilbert space; Spectrum; Orthogonality relations; Fourier expansion; ITERATED FUNCTION SYSTEMS; HARMONIC-ANALYSIS; FRACTAL MEASURES; BERNOULLI CONVOLUTIONS; SPECTRAL THEORY; PERIODICITY; DIMENSIONS; CONJECTURE; DOMAINS; SETS; Mathematics

Abstract

We are concerned with an harmonic analysis in Hilbert spaces L-2(mu), where mu is a probability measure on R-n. The unifying question is the presence of families of orthogonal (complex) exponentials e(lambda)(x) = exp(2 pi i lambda x) in L-2(mu). This question in turn is connected to the existence of a natural embedding of L-2(mu) into an L-2-space of Bohr almost periodic functions on R-n. In particular we explore when L-2(mu) contains an orthogonal basis of e(lambda) functions, for lambda in a suitable discrete subset in R-n; i.e, when the measure mu is spectral. We give a new characterization of finite spectral sets in terms of the existence of a group of local translation. We also consider measures mu that arise as fixed points (in the sense of Hutchinson) of iterated function systems (IFSs), and we specialize to the case when the function system in the IFS consists of affine and contractive mappings in R-n. We show in this case that if mu is then assumed spectral then its partitions induced by the IFS at hand have zero overlap measured in mu. This solves part of the Laba-Wang conjecture. As an application of the new non-overlap result, we solve the spectral-pair problem for Bernoulli convolutions advancing in this way a theorem of Ka-Sing Lau. In addition we present a new perspective on spectral measures and orthogonal Fourier exponentials via the Bohr compactification. Published by Elsevier Inc.

Journal Title

Journal of Functional Analysis

Volume

257

Issue/Number

9

Publication Date

1-1-2009

Document Type

Article

Language

English

First Page

2999

Last Page

3019

WOS Identifier

WOS:000273386900011

ISSN

0022-1236

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