Journal article
Families of Spectral Sets for Bernoulli Convolutions
Journal of Fourier Analysis and Applications, Vol.17(3), pp.431-456
06/2011
DOI: 10.1007/s00041-010-9158-x
Abstract
We study the harmonic analysis of Bernoulli measures μ λ , a one-parameter family of compactly supported Borel probability measures on the real line. The parameter λ is a fixed number in the open interval (0,1). The measures μ λ may be understood in any one of the following three equivalent ways: as infinite convolution measures of a two-point probability distribution; as the distribution of a random power series; or as an iterated function system (IFS) equilibrium measure determined by the two transformations λ(x±1). For a given λ, we consider the harmonic analysis in the sense of Fourier series in the Hilbert space L 2(μ λ ). For L 2(μ λ ) to have infinite families of orthogonal complex exponential functions e 2πis(⋅), it is known that λ must be a rational number of the form $\frac{m}{2n}$ , where m is odd. We show that $L^{2}(\mu_{\frac{1}{2n}})$ has a variety of Fourier bases; i.e. orthonormal bases of exponential functions. For some other rational values of λ, we exhibit maximal Fourier families that are not orthonormal bases.
Details
- Title: Subtitle
- Families of Spectral Sets for Bernoulli Convolutions
- Creators
- Palle Jorgensen - Department of Mathematics University of Iowa Iowa City IA 52242 USAKeri Kornelson - Department of Mathematics University of Oklahoma Norman OK 73019 USAKaren Shuman - Department of Mathematics & Statistics Grinnell College Grinnell IA 50112 USA
- Resource Type
- Journal article
- Publication Details
- Journal of Fourier Analysis and Applications, Vol.17(3), pp.431-456
- DOI
- 10.1007/s00041-010-9158-x
- ISSN
- 1069-5869
- eISSN
- 1531-5851
- Publisher
- SP Birkhäuser Verlag Boston; Boston
- Language
- English
- Date published
- 06/2011
- Academic Unit
- Mathematics
- Record Identifier
- 9983985924402771
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