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Families of Spectral Sets for Bernoulli Convolutions
Authors:Palle E. T. Jorgensen  Keri A. Kornelson  Karen L. Shuman
Affiliation:1. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA
2. Department of Mathematics, University of Oklahoma, Norman, OK, 73019, USA
3. Department of Mathematics & Statistics, Grinnell College, Grinnell, IA, 50112, USA
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 fracm2nfrac{m}{2n}, where m is odd. We show that L2(mfrac12n)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.
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