We prove the almost everywhere convergence of the Marcinkiewicz means of integrable functions σₙf → f for every f ∈ L¹(I²), where I is the group of 2-adic integers.

We apply a construction of generalized twisted convolution to investigate almost everywhere summability of expansions with respect to the orthonormal system of functions ${\ell }_n^a\left(x\right)={(n!/\Gamma (n+a+1))}^{1/2}{e}^{-x/2}{L}_n^a\left(x\right)$, n = 0,1,2,..., in $L^2({ℝ}_{+},x^{a}dx)$, a ≥ 0. We prove that the Cesàro means of order δ > a + 2/3 of any function $f\in L^p\left(x^{a}dx\right)$, 1 ≤ p ≤ ∞, converge to f a.e. The main tool we use is a Hardy-Littlewood type maximal operator associated with a generalized Euclidean convolution.

Using methods from [9] we prove the almost everywhere convergence of the Cesàro means of Laguerre series associated with the system of Laguerre functions $L_n^a\left(x\right)={(n!/\Gamma (n+a+1))}^{1/2}{e}^{-x/2}{x}^{a/2}{L}_n^a\left(x\right)$, n = 0,1,2,..., a ≥ 0. The novel ingredient we add to our previous technique is the $A_p$ weights theory. We also take the opportunity to comment and slightly improve on our results from [9].

We present a method for constructing almost periodic sequences and functions with values in a metric space. Applying this method, we find almost periodic sequences and functions with prescribed values. Especially, for any totally bounded countable set $X$ in a metric space, it is proved the existence of an almost periodic sequence ${\left\{{\psi }_k\right\}}_{k\in \mathbb{Z}}$ such that $\{{\psi }_k;k\in \mathbb{Z}\}=X$ and ${\psi }_k={\psi }_{k+lq\left(k\right)}$, $l\in \mathbb{Z}$ for all $k$ and some $q\left(k\right)\in \mathbb{N}$ which depends on $k$.

The paper is the extension of the author's previous papers and solves more complicated problems. Almost periodic solutions of a certain type of almost periodic linear or quasilinear systems of neutral differential equations are dealt with.

This paper is a continuation of my previous paper in Mathematica Bohemica and solves the same problem but by means of another method. It deals with almost periodic solutions of a certain type of almost periodic systems of differential equations.

This paper generalizes earlier author's results where the linear and quasilinear equations with constant coefficients were treated. Here the method of limit passages and a fixed-point theorem is used for the linear and quasilinear equations with almost periodic coefficients.

We prove an extension of a result by Peres and Solomyak on almost sure absolute continuity in a class of symmetric Bernoulli convolutions.

Almost-periodic solutions in various metrics (Stepanov, Weyl, Besicovitch) of higher-order differential equations with a nonlinear Lipschitz-continuous restoring term are investigated. The main emphasis is focused on a Lipschitz constant which is the same as for uniformly almost-periodic solutions treated in [A1] and much better than those from our investigations for differential systems in [A2], [A3], [AB], [ABL], [AK]. The upper estimates of $\epsilon$ for $\epsilon$-almost-periods of solutions and their derivatives...