We consider the linear elliptic equation −div(a∇u) = f on some bounded domain D, where a has the form a = exp(b) with b a random function defined as b(y) = ∑ j≥1 yjψj where y = (yj) ∈ RN are i.i.d. standard scalar Gaussian variables and (ψj)j≥1 is a given sequence of functions in L∞(D). We study the summability properties of Hermite-type expansions of the solution map y 7→ u(y) ∈ V := H 0 (D), ...

The σ -convergence and σ -core of a real bounded sequence were introduced in [R. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J. 30 (1963) 81–94] and [S.L. Mishra, B. Satapathy, N. Rath, Invariant means and σ -core, J. Indian Math. Soc. 60 (1984) 151–158], respectively. In this work, we extend these ideas to double sequences. c © 2006 Elsevier Ltd. All rights r...

Let e¿ = (S„)jl i, A = (ei)4" j and let A be an infinite matrix which maps E into E where E is an FK-spa.ce with Schauder basis A. Necessary and sufficient conditions in terms of the matrix A are obtained for E to be dense in the summability space Ea = {3c|j43cG£} and conditions are obtained to guarantee that Ea has Schauder basis A. Finally it is shown that if weak and strong sequential conver...

We will examine double sequence to double sequence transformation of independent identically distribution random variables with respect to four-dimensional summability matrix methods. The main goal of this paper is the presentation of the following theorem. If maxk,l|am,n,k,l| maxk,l|am,kan,l| O m−γ1 O n−γ2 , γ1, γ2 > 0, then E|X̆|1 1/γ1 < ∞ and E| ̆̆ X|1 1/γ2 < ∞ imply that Ym,n → μ almost sure P...

We extended a theorem of Mishra and Srivastava (1983–1984) on |C,1|k summability factors, using almost increasing sequences, to |N̄,pn|k summability under weaker conditions.

Some recent results on a general summability method, on the so-called θ-summability is summarized. New spaces, such as Wiener amalgams, Feichtinger’s algebra and modulation spaces are investigated in summability theory. Sufficient and necessary conditions are given for the norm and a.e. convergence of the θ-means.

Let t be a sequence in (0,1) that converges to 1, and define the Abel-type matrix Aα,t by ank = ( k+α k ) tk+1 n (1−tn)α+1 for α>−1. The matrix Aα,t determines a sequenceto-sequence variant of the Abel-type power series method of summability introduced by Borwein in [1]. The purpose of this paper is to study these matrices as mappings into . Necessary and sufficient conditions for Aα,t to be , ...

Let A = (a jn) be an infinite summability matrix. For a given sequence x := (xn), the A-transform of x, denoted by Ax := ((Ax) j), is given by (Ax) j = ∑n=1 a jnxn provided the series converges for each j ∈ N, the set of all natural numbers. We say that A is regular if limAx = L whenever limx = L [4]. Assume that A is a non-negative regular summability matrix. Then x = (xn) is said to be A-stat...

In this paper, we first introduce the notion of summability of an infinite set of vectors of real Hilbert space, without using index sets. Further we introduce the notion of weak summability, which is weaker than that of summability. Then, several statements for summable sets and weakly summable ones are proved. In the last part of the paper, we give a necessary and sufficient condition for sum...

Recently, for single series, the necessary and sufficient conditions $\left\vert C,0\right\vert\Rightarrow \left\vert A_{f}\right\vert_{k}$ vise versa, A_{f}\right\vert \Rightarrow C,0\right\vert_{k}$ versa have been established, where $1 &lt; k \infty $ $A$ is a factorable matrix. The present study extends these results to double summability, also provides some new results.