Nous abordons dans cet article la question de la sommation effective d’une somme de Borel d’une série par la série de factorielles associée. Notre approche fournit un contrôle de l’erreur entre la somme de Borel recherchée et les sommes partielles de la série de factorielles. Nous généralisons ensuite cette méthode au cadre des séries de puissances fractionnaires, après avoir démontré un analogue d’un théorème de Nevanlinna de sommation de Borel fine pour ce cadre.

In this paper we extend the notion of quasinormal convergence via ideals and consider the notion of $ℐ$-quasinormal convergence. We then introduce the notion of $ℐQN\left(ℐwQN\right)$ space as a topological space in which every sequence of continuous real valued functions pointwise converging to $0$, is also $ℐ$-quasinormally convergent to $0$ (has a subsequence which is $ℐ$-quasinormally convergent to $0$) and make certain observations on those spaces.

Let U be a trigonometrically well-bounded operator on a Banach space , and denote by ${ₙ\left(U\right)}_{n=1}^{\infty }$ the sequence of (C,2) weighted discrete ergodic averages of U, that is, $ₙ\left(U\right)=1/n{\sum }_{0<\left|k\right|\le n}(1-|k|/(n+1))U^k$. We show that this sequence ${ₙ\left(U\right)}_{n=1}^{\infty }$ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is x ∈ : Ux = x, and whose null space is the closure of (I - U). This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral theory and...

Writing $(T_R^{\lambda }f)\widehat{}\left(\xi \right)=(1-|\xi {|}^2/R^2{)}_{+}^{\lambda }\widehat{f}\left(\xi \right)$. E. Stein conjectured$$\parallel {\left(\sum _j|T_{R_j}^{\lambda }{f}_i{|}^2\right)}^{1/2}{\parallel }_p\le C\parallel {\left(\sum _j|f_j{|}^2\right)}^{1/2}{\parallel }_p$$for $\lambda \&gt;0$, $\frac{4}{3}\le p\le 4$ and $C=C_{\lambda ,p}$. We prove this conjecture. We prove also $f\left(x\right)={lim}_{j\to \infty }{T}_{2^j}^{\lambda }f\left(x\right)$ a.e. We only assume $\frac{4}{3+2\lambda }\&lt;p\&lt;\frac{4}{1-2\lambda }$.

In this paper we study the set of statistical cluster points of sequences in $m$-dimensional spaces. We show that some properties of the set of statistical cluster points of the real number sequences remain in force for the sequences in $m$-dimensional spaces too. We also define a notion of $\Gamma$-statistical convergence. A sequence $x$ is $\Gamma$-statistically convergent to a set $C$ if $C$ is a minimal closed set such that for every $ϵ>0$ the set $\{k\rho (C,x_k)\ge ϵ\}$ has density zero. It is shown that every statistically bounded sequence...

In this paper the ideas of three types of statistical convergence of a sequence of random variables, namely, statistical convergence in probability, statistical convergence in mean of order $r$ and statistical convergence in distribution are introduced and the interrelation among them is investigated. Also their certain basic properties are studied.

In this paper we use the notion of statistical convergence of infinite series naturally introduced as the statistical convergence of the sequence of the partial sums of the series. We will discuss some questions related to the convergence of subseries of a given series.

We study several kinds of statistical convergence of sequences of functions with values in semi-uniform spaces. Particularly, we generalize to statistical convergence the classical results of C. Arzelà, Dini and P.S. Alexandroff, as well as their statistical versions studied in [Caserta A., Di Maio G., Kočinac L.D.R., {Statistical convergence in function spaces},. Abstr. Appl. Anal. 2011, Art. ID 420419, 11 pp.] and [Caserta A., Kočinac L.D.R., {On statistical exhaustiveness}, Appl. Math. Lett....

This paper is closely related to the paper of Harry I. Miller: Measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995), 1811–1819 and contains a general investigation of statistical convergence of subsequences of an arbitrary sequence from the point of view of Lebesgue measure, Hausdorff dimensions and Baire’s categories.

Schmidt’s classical Tauberian theorem says that if a sequence $(s_k:k=0,1,...)$ of real numbers is summable (C,1) to a finite limit and slowly decreasing, then it converges to the same limit. In this paper, we prove a nondiscrete version of Schmidt’s theorem in the setting of statistical summability (C,1) of real-valued functions that are slowly decreasing on ℝ ₊. We prove another Tauberian theorem in the case of complex-valued functions that are slowly oscillating on ℝ ₊. In the proofs we make use of two nondiscrete...