Related papers: Averaging almost periodic functions along exponent…

Given an observation of the uniform empirical process $\alp_n$, its functional increments $\alp_n(u+a_n\cdot)-\alp_n(u)$ can be viewed as a single random process, when $u$ is distributed under the Lebesgue measure. We investigate the almost…

We study the continuity properties of trajectories for some random series of functions $\sum a\_kf(\alpha X\_k(\omega))$ where $a\_k$ is a complex sequence, $X\_k$ a sequence of real independent random variables, $f$ is a real valued…

For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.

For an ergodic map $T$ and a non-constant, real-valued $f \in L^1$, the ergodic averages $\mathbb{A}_N f(x) = \frac{1} {N} \sum_{n=1}^N f(T^n x)$ converge a.e., but the convergence is never monotone. Depending on particular properties of…

We provide an introduction of some basic facts of uniformly almost periodic functions, such as Fourier series representations. A result is then proved about Fourier coefficients which is a generalization of the purely periodic case. We then…

Denote by {$\times$} the fractional part. We establish several new metrical results on the distribution properties of the sequence ({x n }) n$\ge$1. Many of them are presented in a more general framework, in which the sequence of functions…

A classical fact of the theory of almost periodic functions is the existence of their asymptotic distributions. In probabilistic terms, this means that if $f$ is a Besicovitch almost periodic function and $V$ is a random variable uniformly…

We investigate the probability of observing a given pattern of $n$ rises and falls in a random stationary data series. The data are modelled as a sequence of $n+1$ independent and identically distributed random numbers. This probabilistic…

Given an arbitrary long but finite sequence of observations from a finite set, we construct a simple process that approximates the sequence, in the sense that with high probability the empirical frequency, as well as the empirical one-step…

We consider certain finite sets of circle-valued functions defined on intervals of real numbers and estimate how large the intervals must be for the values of these functions to be uniformly distributed in an approximate way. This is used…

A general statistical thermodynamic theory that considers given sequences of x-integers to play the role of particles of known type in an isolated elastic system is proposed. By also considering some explicit discrete probability…

Let $(f_n)_{n=1}^{\infty}$ be a sequence of polynomials and $\alpha>1$. In this paper we study the distribution of the sequence $(f_n(\alpha))_{n=1}^{\infty}$ modulo one. We give sufficient conditions for a sequence $(f_n)_{n=1}^{\infty}$…

In some particular cases we give criteria for morphic sequences to be almost periodic (=uniformly recurrent). Namely, we deal with fixed points of non-erasing morphisms and with automatic sequences. In both cases a polynomial-time algorithm…

In this short note we study uniform approximations to the normal distributions by Jacobi theta functions. We shall show that scaled theta functions approach to a normal distribution exponentially fast.

This is an attempt of a comprehensive survey of the results in which estimates of the norms of linear means of multiple Fourier series, the Lebesgue constants, are obtained by means of estimating the Fourier transform of a function…

Assume that $\alpha>1$ is an algebraic number and $\xi\neq0$ is a real number. We are concerned with the distribution of the fractional parts of the sequence $(\xi \alpha^{n})$. Under various Diophantine conditions on $\xi$ and $\alpha$, we…

The aim of this paper is to introduce and to study an algebra of almost periodic generalized functions containing the classical Bohr almost periodic functions as well as almost periodic Schwartz distributions

We study intermediate-scale statistics for the fractional parts of the sequence $(\alpha a_n)_{n=1}^{\infty}$, where $(a_n)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and $\alpha\in\mathbb{R}$. In particular, we consider…

A review of the state of the art of the comparison between any two different modes of convergence of sequences of measurable functions is carried out with focus on the algebraic structure of the families under analysis. As a complement of…

On the sets of $2\pi$-periodic functions $f$, which are defined with a help of $(\psi, \beta)$-integrals of the functions $\varphi$ from $L_{1}$, we establish Lebesgue-type inequalities, in which the uniform norms of deviations of Fourier…