Families of Spectral Sets for Bernoulli Convolutions |
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Authors: | Palle E T Jorgensen Keri A Kornelson Karen L Shuman |
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Institution: | 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
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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
\fracm2n\frac{m}{2n}, where m is odd. We show that
L2(m\frac12n)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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Keywords: | |
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