Related papers: Vilenkin-Fourier series in variable Lebesgue space…

We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs.…

In this note, we introduce the notion of modulus of $p$-variation for a function of a real variable, and show that it serves in at least two important problems, namely, the uniform convergence of Fourier series and computation of certain…

In this paper we investigate some convergence and divergence of some specific subsequences of partial sums with respect to Walsh system on the martingale Hardy spaces. By using these results we obtain relationship of the ratio of…

Let $f$ be a locally integrable function defined on $\mathbb{R}$, and let $(n_k)$ be a lacunary sequence. Define the operator $A_{n_k}$ by $$A_{n_k}f(x)=\frac{1}{n_k}\int_0^{n_k}f(x-t)\, dt.$$ We prove various types of new inequalities for…

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $\mathfrak g$ of left-invariant vector fields on a Lie group $\mathbb G$ and…

This paper is concerned with the study of the fractional finite sums theory. We present the classes of functions for which it is possible to characterize the constant related to the derivative of fractional sums (denominated by essence of a…

We study $\mathbb{R}_{\textrm{an},\exp}$-definable functions $f:\mathbb{R}\to \mathbb{R}$ that take integer values at all sufficiently large positive integers. If $|f(x)|= O\big(2^{(1+10^{-5})x}\big)$, then we find polynomials $P_1, P_2$…

We show the sum of the first $k$ Lyapunov exponents of linear cocycles is an upper semicontinuous function in the $L^p$ topologies, for any $1 \le p \le \infty$ and $k$. This fact, together with a result from Arnold and Cong, implies that…

Indicator functions mentioned in the title are constructed on an arbitrary nondiscrete locally compact Abelian group of finite dimension. Moreover, they can be obtained by small perturbation from any indicator function fixed beforehand. In…

We obtain results concerning the so-called factorization for the convergence of random variables almost everywhere (almost surely or with probability one), belonging to the classical Lebesgue-Riesz spaces and we extend these results to the…

In this paper we shall be concerned with $H_\alpha$ summability, for $0 < \alpha \leq 2$ of the Fourier series of arbitrary $L^1([-\pi,\pi]) $ functions. The methods to be employed here are a refinement of the real variable methods…

In an effort to extend classical Fourier theory, Hedenmalm and Montes-Rodr\'{\i}guez (2011) found that the function system \[ e_m(x)=e^{i\pi mx},\quad e_n^\dagger(x)=e_n(-1/x)=e^{-i\pi n/x} \] is weak-star complete in…

Given any non-polynomial $G$-function $F(z)=\sum_{k=0}^\infty A_k z^k$ of radius of convergence $R$ and in the kernel a $G$-operator $L_F$, we consider the $G$-functions $F_n^{[s]}(z)=\sum_{k=0}^\infty \frac{A_k}{(k+n)^s}z^k$ for every…

We show the pointwise convergence of the averages \[ \mathcal{A}_N f(x) = \frac{1}{\# \mathbf{B}_N} \sum_{n \in \mathbf{B}_N} f(x + n) \] for $f \in \ell^1(\mathbb{Z})$ where $\mathbf{B}_N = \mathbf{B} \cap [1, N]$, and $\mathbf{B}$ is a…

We construct a $1$-bounded completely multiplicative function $f$ whose logarithmically-averaged partial sums satisfy $$ \limsup_{x \rightarrow \infty} \frac{\left|\sum_{n \leq x} \frac{f(n)}{n}\right|}{1+\exp\left(\sum_{p \leq x}…

We say that a function $f:[0,1]\rightarrow \R$ is \emph{nowhere $L^q$} if, for each nonvoid open subset $U$ of $[0,1]$, the restriction $f|_U$ is not in $L^q(U)$. For a fixed $1 \leq p <\infty$, we will show that the set $$ S_p\doteq {f \in…

Let $1<p,q<\infty ,\ \theta_1 \geq 0,\ \theta_2 \geq 0$ and let $a(x), b(x)$ be a weight functions. In the present paper we intend to study the function space $A_{q),\theta _{2}}^{p),\theta _{1}}\left( \mathbb R^n\right)$ consisting of all…

Let $f_k$ be the $k$-th Fourier coefficient of a function $f$ in terms of the orthonormal Hermite, Laguerre or Jacobi polynomials. We give necessary and sufficient conditions on $f$ for the inequality $\sum_{k}|f_k|^2\theta^k<\infty$ to…

Suppose $1 < p < \infty$. Carleson's Theorem states that the Fourier series of any function in $L^p[-\pi, \pi]$ converges almost everywhere. We show that the Schnorr random points are precisely those that satisfy this theorem for every $f…

Motivated by applications in number theory, analysis, and fractal geometry, we consider regularity properties and dimensions of graphs associated with Fourier series of the form $F(t)=\sum_{n=1}^\infty f(n)e^{2\pi i nt}/n$, for a large…