Absolute continuity of complex Bernoulli convolutions

Journal of Fourier Analysis and Applications, 2011

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.

2000

Abstract. The distribution v\ of the random series Y,±A" is the infinite convolution product of|(6-A"+ 6\ n). These measures have been studied since the 1930s, revealing connections with harmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorff dimension estimation. In this survey we describe some of these connections, and the progress that has been made so far on the fundamental open problem: For which A€(i, 1) is v\ absolutely continuous?

Acta Mathematica Hungarica, 2001

Let O be an open set in the complex plane and let ? be a holomorphic function on O. Let K be a compact subset of O with nonempty interior such that 0 ? ?K. Let µ be the Borel measure of R4 ? C2 given by

Annals of Probability an Official Journal of the Institute of Mathematical Statistics, 2007

We prove a multivariable approximate Carleman theorem on the determination of complex measures on ${\mathbb{R}}^n$ and ${\mathbb{R}}^n_+$ by their moments. This is achieved by means of a multivariable Denjoy--Carleman maximum principle for quasi-analytic functions of several variables. As an application, we obtain a discrete Phragm\'{e}n--Lindel\"{o}f-type theorem for analytic functions on ${\mathbb{C}}_+^n$.

Acta Mathematica Hungarica, 1996

Proceedings of the American Mathematical …, 1996

We prove for every non-abelian compact connected group G there is a continuous, singular, central measure µ with µ * µ in L p for all p, 1 ≤ p < ∞. We also construct such measures on some families of non-abelian compact totally disconnected groups. These results settle an open question of Ragozin.

The purpose of this paper is to prove several results in approximations through complex convolution polynomials with Jackson-type rate or with best approximation rate, having the quality of preservation of some properties in geometric function theory, like the preservation of: coefficients' bounds, positive real part, bounded turn, close-to-convexity, starlikeness, convexity, spirallikeness, α-convexity. Also, some sufficient conditions for starlikeness and univalence of analytic functions are preserved.

2018

A theorem of Godefroy and Shapiro states that non-trivial convolution operators on the space of entire functions on C are hypercyclic. Moreover, it was shown by Bonilla and Grosse-Erdmann that they have frequently hypercyclic functions of exponential growth. On the other hand, in the infinite dimensional setting, the Godefroy-Shapiro theorem has been extended to several spaces of entire functions defined on Banach spaces. We prove that on all these spaces, non-trivial convolution operators are strongly mixing with respect to a gaussian probability measure of full support. For the proof we combine the results previously mentioned and we use techniques recently developed by Bayart and Matheron. We also obtain the existence of frequently hypercyclic entire functions of exponential growth.

Preprint CUHK, 2002

Bernoulli convolutions (= infinitely convolved Bernoulli measures) have been studied extensively since the early 1930's , later in the 1960's and more recently, in the 1990's onwards, following the work of Alexander and Yorke [1]. They have also been considered in view of their applications to fractal geometry and ergodic theory (see for further details and references). In the present paper we follow the approach based on multifractal analysis (w.r.t. the local dimension) developed in , where the classical Legendre transform formula is proved for the weak Gibbs measures in the sense of Yuri . Our motivation comes from the fact that the Bernoulli convolution associated with the golden ratio (the Erdős measure) proves to be weak Gibbs, and in the present paper we aim to generalize this result.

J. Inequal. Pure Appl. Math, 2002

For functions belonging to each of the subclasses M * n (α) and N * n (α) of normalized analytic functions in open unit disk U, which are introduced and investigated in this paper, the authors derive several properties involving their generalized convolution by applying certain techniques based especially upon the Cauchy-Schwarz and Hölder inequalities. A number of interesting consequences of these generalized convolution properties are also considered.