Related papers: On the relation between Lebesgue summability and s…

Assume that a convergent series of real numbers $\sum\limits_{n=1}^\infty a_n$ has the property that there exists a set $A\subseteq \N$ such that the series $\sum\limits_{n \in A} a_n$ is conditionally convergent. We prove that for a given…

Given a triangular array $\left\{X_{n,k}, \, 1 \leqslant k \leqslant n, n \geqslant 1 \right\}$ of random variables satisfying $\mathbb{E} \lvert X_{n,k} \rvert^{p} < \infty$ for some $p \geqslant 1$ and sequences $\{b_{n} \}$, $\{c_{n} \}$…

For an array $\left\{X_{n,j}, \, 1 \leqslant j \leqslant k_{n}, n \geqslant 1 \right\}$ of random variables and a sequence $\{c_{n} \}$ of positive numbers, sufficient conditions are given under which, for all $\varepsilon > 0$,…

We draw attention to simplifications in the theory of a N\"orlund summation method $(N, p)$ that arise when the series $\sum_{n \geqslant 0} p_n$ is convergent.

The main purpose of this paper is to introduce the concepts of Wijsman $C_{\lambda}$ statistical convergence, Wijsman $C_{\lambda}$ summability and Wijsman $\mathcal{I}$-$C_{\lambda}$ summability for sequence of sets by using submethod.…

For the functions from sets $C_\beta^\psi C$ and $C_\beta^\psi L_s, \ 1\leq s\leq\infty$, generated by sequences $\psi(k)>0$ satisfying the condition d'Alembert $\mathop {\rm \lim}\limits_{k\rightarrow\infty}\frac{\psi(k+1)}{\psi(k)}=q, \…

We find an equivalent condition for a real function $f:[a,b]\to\R$ to be Lebesgue equivalent to an $n$-times differentiable function ($n\geq 2$); a simple solution in the case $n=2$ appeared in an earlier paper. For that purpose, we…

In this paper, we introduce 3-dimensional $L-$summing method, which is a rearrangement of the summation $\sum A_{abc}$ with $1\leq a,b,c\leq n$. Applying this method on some special arrays, we obtain some identities on the Riemann zeta…

We investigate the connection between the bubble-resummation and critical-point methods for computing the $\beta$-functions in the limit of large number of flavours, $N$, and show that these can provide complementary information. While the…

We introduce a certain class of summability methods on $L^{\infty}([1, \infty))$ which is defined by convolution in the group algebra $L^1(\mathbb{R}^{\times})$. This class contains an integral version of Ces\`{a}ro summability method and…

In this note we give a short and self-contained proof that, for any $\delta > 0$, $\sum_{x \leq n \leq x+x^\delta} \lambda(n) = o(x^\delta)$ for almost all $x \in [X, 2X]$. We also sketch a proof of a generalization of such a result to…

Let $\zeta(.)$ denote the Riemann zeta function and let $a(.)$ and $A(.)$ respectively denote a multiplicative function and its corresponding summatory function. We consider the correlation $$ \langle a(n)A(n-1) \rangle (T) =…

A sequence of real numbers $\{x_{n}\}_{n\in \mathbb{N}}$ is said to be $\alpha \beta$-statistically convergent of order $\gamma$ (where $0<\gamma\leq 1$) to a real number $x$ \cite{a} if for every $\delta>0,$ $$\underset{n\rightarrow…

A well known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In this paper we give sufficient conditions for an analogue of this theorem to hold for…

We consider nonlinear, or "event-dependent", sampling, i.e. such that the sampling instances {tk} depend on the function being sampled. The use of such sampling in the construction of Lebesgue's integral sums is noted and discussed as…

We study the sum $\sum_{abc \leq x} \Omega([a,b,c])$, where $\Omega(n)$ denotes the number of distinct prime divisors of $n \in \mathbb{Z}_{\geq 1}$, counted with multiplicity, and where $(a,b,c) = \gcd(a,b,c)$ and $[a,b,c] =…

We prove that there exist positive constants $C$ and $c$ such that for any integer $d \ge 2$ the set of ${\mathbf x}\in [0,1)^d$ satisfying $$ cN^{1/2}\le \left|\sum^N_{n=1}\exp\left (2 \pi i \left (x_1n+\ldots+x_d n^d\right)\right)…

This article is a study on the summability of random Fourier--Jacobi series of some functions in different spaces. We consider the random series $ \sum_{n=0}^\infty a_nA_n(\omega)p_n^{(\gamma,\delta)}(y), $ where…

We provide a strengthened version of the famous Jakobson's theorem. Consider an interval map $f$ satisfying a summability condition. For a generic one-parameter family $f_t$ of maps with $f_0=f$, we prove that $t=0$ is a Lebesgue density…

Suppose a complex function $f$ has a Lebesgue measurable inverse Laplace transform. We show that the $n$th order forward and backward differences of $f$ at $z_0\in\mathbb{C}$ tend to zero as $n\to\infty$ whenever $z_0$ lies in the region of…