Convergence Of Double Series Research Articles

$$AK(\vartheta )$$-Property of Double Series Spaces

In the present paper, we show that $$\vartheta $$ -convergent double series spaces $${\mathcal {CS}}_\vartheta $$ and the series space $$\mathcal {BV}$$ of double sequences of bounded variation are BDK-spaces and investigate their $$AK(\vartheta )$$ -space property, where $$\vartheta \in \{p,bp,r\}$$ .

On almost certain convergence of double series of random elements and the rate of convergence of tail series

ABSTRACT For a double array of random elements taking values in a real separable Banach space, we provide sufficient conditions for the double series to converge almost certainly to a random element S as . For a convergent double series, we study the rate of convergence to a random element S by studying the rate at which the corresponding tail series converges almost certainly to 0 as where .

Uniqueness Theorems for Franklin Series

Simple Franklin series are investigated that converge to zero everywhere except for one point (or several points). It is also proved that the one-point set (or a finite set) is a uniqueness set for Pringsheim convergent double series.

A summation due to Ramanujan revisited

We employ Hardy’s regularly convergent double series to refine an argu- ment of Nanjundiah [7]. In particular, we evaluate some alternating series.

A singularly perturbed Neumann problem for the Poisson equation in a periodically perforated domain. A functional analytic approach

We consider a Neumann problem for the Poisson equation in the periodically perforated Euclidean space. Each periodic perforation has a size proportional to a positive parameter ε. For each positive and small ε, we denote by a suitably normalized solution. Then we are interested to analyze the behavior of when ε is close to the degenerate value , where the holes collapse to points. In particular we prove that if , then can be expanded into a convergent series expansion of powers of ε and that if then can be expanded into a convergent double series expansion of powers of ε and . Our approach is based on potential theory and functional analysis and is alternative to those of asymptotic analysis.

Free-space communications over exponentiated Weibull turbulence channels with nonzero boresight pointing errors.

In this paper, we present analytical expressions for the performance of urban free-space optical (FSO) communication systems under the combined influence of atmospheric turbulence- and misalignment-induced fading (pointing errors). The atmospheric turbulence channel is modeled by the exponentiated Weibull (EW) distribution that can accurately describe the probability density function (PDF) of the irradiance fluctuations associated with a transmitted Gaussian-beam wave and a finite-sized receiving aperture. The nonzero boresight pointing error PDF model, which is recently proposed for considering the effects of both boresight and jitter, is adopted in analysis. We derive a novel expression for the composite PDF in terms of a convergent double series involving a Meijer's G-function. Based on the statistical results mentioned above, exact expressions for the average bit error rate of on-off keying modulation scheme and the outage probability are developed. To provide more insight, we also perform an asymptotic error rate analysis at high average signal-to-noise ratio. Our analytical results indicate that the diversity gain for the zero boresight case is determined only by the ratio between the equivalent beamwidth at the receiver and the jitter standard deviation, while for the nonzero boresight case, the diversity gain is related to the ratio of the equivalent beamwidth to the jitter variance as well as the parameter of the EW distribution.

A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach

We consider a heat transmission problem for a composite material which fills the $n$-dimensional Euclidean space. The composite has a periodic structure and consists of two materials. In each periodicity cell one material occupies a cavity of size $\epsilon$, and the second material fills the remaining part of the cell. We assume that the thermal conductivities of the materials depend nonlinearly upon the temperature. We show that for $\epsilon$ small enough the problem has a solution, \textit{i.e.}, a pair of functions which determine the temperature distribution in the two materials. Then we analyze the behavior of such a solution as $\epsilon$ approaches $0$ by an approach which is alternative to those of asymptotic analysis. In particular we prove that if $n\geq 3$, the temperature can be expanded into a convergent series expansion of powers of $\epsilon$ and that if $n=2$ the temperature can be expanded into a convergent double series expansion of powers of $\epsilon$ and $\epsilon \log \epsilon$.

Almost Conservative Four-Dimensional Matrices through de la Vallée-Poussin Mean

The purpose of this paper is to generalize the concept of almost convergence for double sequence through the notion of de la Vallée-Poussin mean for double sequences. We also define and characterize the generalized regularly almost conservative and almost coercive four-dimensional matrices. Further, we characterize the infinite matrices which transform the sequence belonging to the space of absolutely convergent double series into the space of generalized almost convergence.

A note on the series representation for the density of the supremum of a stable process

An absolutely convergent double series representation for the density of the supremum of $\alpha$-stable Lévy process was obtained by Hubalek and Kuznetsov for almost all irrational $\alpha$. This result cannot be made stronger in the following sense: the series does not converge absolutely when $\alpha$ belongs to a certain subset of irrational numbers of Lebesgue measure zero. Our main result in this note shows that for every irrational $\alpha$ there is a way to rearrange the terms of the double series, so that it converges to the density of the supremum. We show how one can establish this stronger result by introducing a simple yet non-trivial modification in the original proof of Hubalek and Kuznetsov.

Absolute convergence of double series of Fourier-Haar coefficients for functions of bounded p-variation

We consider functions of two variables of bounded p-variation of the Hardy type on the unit square. For these functions we obtain a sufficient condition for the absolute convergence of series of positive powers of Fourier coefficients with power-type weights with respect to the double Haar system. This condition implies those for the absolute convergence of series of Fourier-Haar coefficients of one-variable functions which have a bounded Wiener p-variation or belong to the class Lip α. We show that the obtained results are unimprovable. We also formulate N-dimensional analogs of the main result and its corollaries.

On convergent series representations of Mellin-Barnes integrals

Multiple Mellin-Barnes integrals are often used for perturbative calculations in particle physics. In this context, the evaluation of such objects may be performed through residues calculations which lead to their expression as multiple power series and logarithms of the parameters involved in the problem under consideration. However, in most of the cases, several convergent series representations exist for a given integral. These series converge in different regions of values of the parameters, and it is not obvious to obtain them. For twofold integrals, we present a method which allows to derive straightforwardly and systematically: First, different sets of poles which correspond to different convergent double series representations of a given integral, second the regions of convergence of all these series (without an a priori full knowledge of their general term), and third the general term of each series (this may be performed, if necessary, once the relevant domain of convergence has been found). This systematic procedure is illustrated with some integrals which appear, among others, in the calculation of the two-loop hexagon Wilson loop in $\mathcal {N} = 4$N=4 SYM theory. Mellin-Barnes integrals of higher dimension are also considered.

Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole

Let \({\mathbb{A}}(\epsilon)\) be the annular domain obtained by removing from a bounded open domain \({\mathbb{I}}^{o}\) of ℝn a small cavity of size ϵ>0. Then we assume that for some natural index l, \(\lambda_{l}[{\mathbb{I}}^{o}]>0\) is a simple Neumann eigenvalue of −Δ in \({\mathbb{I}}^{o}\), and we show that there exists a real valued real analytic function \(\hat{\lambda }_{l}(\cdot,\cdot)\) defined in an open neighborhood of (0,0) in ℝ2 such that the lth Neumann eigenvalue \(\lambda_{l}[{\mathbb{A}}(\epsilon)]\) of −Δ in \({\mathbb{A}}(\epsilon)\) equals \(\hat{\lambda}_{l}(\epsilon,\kappa_{n}\epsilon\log\epsilon)\) and such that \(\hat{\lambda}_{l}(0,0)= \lambda_{l}[{\mathbb{I}}^{o}]\). Here κn=1 if n is even and κn=0 if n is odd. Thus in particular, we show that if n is even \(\lambda_{l}[{\mathbb {A}}(\epsilon)]\) can be expanded into a convergent double series of powers of ϵ and ϵlogϵ and that if n is odd \(\lambda_{l}[{\mathbb{A}}(\epsilon)]\) can be expanded into a convergent series of powers of ϵ. Then related statements have been proved for corresponding eigenfunctions.

An inclusion theorem for double Cesàro matrices over the space of absolutely [formula omitted]-convergent double series

In this work it is shown that if C ( α , β ) and C ( γ , δ ) are double Cesàro matrices with α ≥ γ > − 1 , β ≥ δ > − 1 , then any double series that is ∣ C ( γ , δ ) ∣ k -summable is also ∣ C ( α , β ) ∣ k -summable, k ≥ 1 .

Series expansions for the third incomplete elliptic integral via partial fraction decompositions

We find convergent double series expansions for Legendre's third incomplete elliptic integral valid in overlapping subdomains of the unit square. Truncated expansions provide asymptotic approximations in the neighborhood of the logarithmic singularity ( 1 , 1 ) if one of the variables approaches this point faster than the other. Each approximation is accompanied by an error bound.

On Convergence of Double Series

For the convergence of double series and iterated series, a sufficient condition is obtained. This result provides a test for the convergence of double series and iterated series.

Absolute, perfect, and unconditional convergence of double series

Absolute, perfect, and unconditional convergence of double series

The Summation of Slowly Convergent Double Series and Application to Fluid Mechanics

The Summation of Slowly Convergent Double Series and Application to Fluid Mechanics

On the rate of convergence of double series of exponents representing regular functions on products of convex polygons

Estimates exact in order are obtained in the uniform and integral metrics for the deviation of partial sums of double series of exponents that represent functions which are regular on products of convex polygons and either continuous on a product of closed polygons or belonging to the Smirnov class.

On a representation of the Green's functions for shallow panels supported on the boundary of a rectangular base

To enhance the convergence of double series in the case of a shallow panel supported on a rectangular base and a long closed cylindrical shell we propose an approximate representation of the Green's function as a combination of rapidly convergent expansions and a closed-form analytic component obtained by approximate summation of one particular part of series using the two-dimensional integral Fourier transform and reduction to the Kelvin functions.

A Note on Slowly Convergent Double Series

A Note on Slowly Convergent Double Series