Families of Spectral Sets for Bernoulli Convolutions

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 m 2n , where m is odd. We show that L 2 (μ 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.

Spectra of a class of self-affine measures

“…We first give some remarks on these questions, and answer a question of Dutkay and Jorgensen [3] in Section 2. We then consider the spectra of Bernoulli convolutions in Section 3, and obtain a sharp result which extends the corresponding result of Jorgensen, Kornelson and Shuman [13]. In the final section, we provide a structural property for the integer spectrum of a spectral self-affine measure μ M,D .…”

“…The other results concerning the spectral property of the Bernoulli convolutions have been obtained in [12,9,7,6,13]. Usually the following two types of questions have often been considered: For what values of λ, is μ λ a spectral measure?…”

“…When it is, what are the Fourier bases in the space L 2 (μ λ )? Recently, Jorgensen, Kornelson and Shuman [13] find a condition on p and k which determines whether E(pΓ ( 1 2k )) is an ONB for L 2 (μ 1 2k ). The main results of [13] are the following.…”

Scaling of spectra of a class of self‐similar measures on R

“…For the spectrality of , Jorgensen and Pedersen first proved that for each , the set given by is a spectrum of . Later, Jorgensen, kornelson and Shuman proved that if such that , then is a spectrum of . In 2011, Li showed that if such that , then is a spectrum of .…”

“…} is a spectrum of 1∕ (2 ) . Later, Jorgensen, kornelson and Shuman [14] proved that if ∈ 2 + 1 such that < 2(2 −1) , then 2 Λ 2 is a spectrum of 1∕ (2 ) . In 2011, Li [19] showed that if ∈ 2 + 1 such that < 2 − 1, then 2 Λ 2 is a spectrum of 1∕ (2 ) .…”