Search

In this paper, we study the Kantorovich-Stancu-type generalization of Szász-Mirakyan operators including Brenke-type polynomials and prove a Korovkin-type theorem via the T -statistical convergence and power series summability method. Moreover, we determine the rate of the convergence. Furthermore, we establish the Voronovskaya- and Grüss-Voronovskaya-type theorems for T -statistical convergence. [ABSTRACT FROM AUTHOR]

In the past, Eshelby’s method for a single inclusion in an infinite medium has been extended to periodic heterogeneous materials for the evaluation of global properties. In this work, the extended Eshelby’s method is used for the evaluation of local fields such as strain and stress in periodic heterogeneous materials characterized by unit cells containing multiple fibres or voids with different geometric and mechanical properties. The proposed method provides Fourier coefficients that are used to construct partial sums of three-dimensional trigonometric Fourier series of local fields. These partial sums exhibit unwanted effects such as the Gibbs phenomenon. In order to attenuate these effects, the behaviour of iterated Fejér partial sums and means of the Riesz summability method is investigated. Extensive numerical examples on both a multiphase composite and a material with voids are provided: in the examples, partial sums, iterated Fejér partial sums, and Riesz means for the local stress are compared with FEM solutions. The numerical comparison shows Riesz means perform better than partial sums and iterated Fejér partial sums and are effective in approximating elastic local fields in periodic heterogeneous solids. [ABSTRACT FROM AUTHOR]

We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form εzf′=Fε,z,f with F a Cν-valued function, holomorphic in a polydisc D-ρ×D-ρ×D-ρν. We show that its unique formal solution in power series of ε, whose coefficients are holomorphic functions of z, is 1-summable under a Siegel-type condition on the eigenvalues of Ff(0,0,0). The estimates employed resemble the ones used in KAM theorem. A simple lemma is applied to tame convolutions that appear in the power series expansion of nonlinear equations. Applications to spherical Bessel functions and probability theory are indicated. The proposed summability method has certain advantages as it may be applied as well to (singularly perturbed) nonlinear partial differential equations of evolution type. [ABSTRACT FROM AUTHOR]

We show a Dvoretzky-Rogers type theorem for the adapted version of the q-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector valued version of convergence in the weak topology, is equivalent to the convergence with respect to the norm. Examples and applications are also given. [ABSTRACT FROM AUTHOR]

We study the Walsh series expansion of multivariate functions in Lp (1 ≤ p ≤ ∞) and, in particular, in Lip(α, p). The rate of uniform approximation by T-transformation of rectangular partial sums of double Walsh to these functions is investigated. By extending the concepts of rest (head) bounded variation series, which was introduced by Leindler (2004), we generalize the related results of Móricz and Rhoades (1996), Nagy (2012). Our results can be applied to many summability methods, including the Norlund summability and weighted summability. [ABSTRACT FROM AUTHOR]

Two concepts--one of statistical convergence and the other of de la Vallée-Poussinmean--play an important role in recent research on summability theory. In this work we define a new type of summability methods and statistical completeness involving the ideas of de la Vallée-Poussin mean and statistical convergence in the framework of probabilistic normed spaces. [ABSTRACT FROM AUTHOR]

Mittal and Rhoades (1999, 2000) and Mittal et al. (2011) have initiated a study of error estimates En(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper, the first author continues the work in the direction for T to be a Np-matrix. We extend two theorems on summability matrix Np of Deger et al. (2012)where they have extended two theorems of Chandra (2002) using Cλ-method obtained by deleting a set of rows from Cesàro matrix C1. Our theorems also generalize two theorems of Leindler (2005) to Np-matrix which in turn generalize the result of Chandra (2002) and Quade (1937). [ABSTRACT FROM AUTHOR]

The idea of I-convergence of real sequences was introduced by Kostyrko et al., (2000/01) and also independently by Nuray and Ruckle (2000). In this paper, we introduce the concepts of (δm, I)-statistical convergence of order ɑ and strong (δpm, I)-Cesàro summability of order ɑ of real sequences and investigated their relationship. [ABSTRACT FROM AUTHOR]

In 2002 Bennett et al. started the investigation to which extent sequence spaces are determined by the sequences of 0s and 1s they that contain. In this relation they defined three types of Hahn properties for sequence spaces: the Hahn property, separable Hahn property, and matrix Hahn property. In general all these three properties are pairwise distinct. If a sequence space E is solid and ({0, 1}N n E)β = Eβ = l¹ then the two last properties coincide. We will show that even on these additional assumptions the separable ahn property and the Hahn property still do not coincide. However if we assume E to be the bounded summability domain of a regular Riesz matrix Rp or a regular nonnegative Hausdorff matrix Hp, then this assumption alone guarantees that E has the Hahn property. For any (infinite) matrix A the Hahn property of its bounded summability domain is related to the strongly nonatomic property of the density dA defined by A. We will find a simple necessary and sufficient condition for the density dA defined by the generalized Riesz matrix Rp,m to be strongly nonatomic. This condition appears also to be sufficient for the bounded summability domain of Rp,m to have the Hahn property. [ABSTRACT FROM AUTHOR]

We intend to make a new approach and introduce the concepts of statistical convergence of order ᾶ and strongly p-Cesàro summability of order ᾶ for double sequences of complex or real numbers. Also, some relations between the statistical convergence of order ᾶ and strong p-Cesàro summability of order ᾶ are given. [ABSTRACT FROM AUTHOR]

We prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical summability (C, 1, 1). We also study the rate of statistical summability (C, 1, 1) of positive linear operators. Finally we construct an example to show that our result is stronger than those previously proved for Pringsheim's convergence and statistical convergence. [ABSTRACT FROM AUTHOR]

A theory of summability of orthonormal sets is introduced in multinormed spaces. The approach which is presented caters for infinite sets {øj}jεj, where the index set may be uncountable, and is applied to obtain convergence results in appropriate spaces of test functions and corresponding spaces of generalized functions.These spaces are constructed in a systematic manner that relies heavily on properties of orthonormal bases in Hilbert spaces. A space of almost-periodic generalized functions, in which each generalized function can be expanded in terms of an uncountable basis of exponential functions, is obtained as a special case of our theory. [ABSTRACT FROM AUTHOR]

We introduce the concept of statistical convergence of order α of difference sequences, and we give some relations between the set of statistical convergence of order α of difference sequences and strong wαγ(△m,p)-summability of order α. Furthermore some relations between the spacewαγ(△m and Sαγ are examined. [ABSTRACT FROM AUTHOR]

We present optimal convergence results for themth derivative of a function by sequences of linear operators. The usual convergence is replaced by A-summability, with A being a sequence of infinite matrices with nonnegative real entries, and the operators are assumed to be m-convex. Saturation results for nonconvergent but almost convergent sequences of operators are stated as corollaries. [ABSTRACT FROM AUTHOR]

Korovkin type approximation theorems are useful tools to check whether a given sequence (Ln)n=1 of positive linear operators on C[0, 1] of all continuous functions on the real interval [0, 1] is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, x, and x2 in the space C[0, 1] as well as for the functions 1, cos, and sin in the space of all continuous 2π-periodic functions on the real line. In this paper, we use the notion of B-statistical A-summability to prove the Korovkin second approximation theorem. We also study the rate of B-statistical A-summability of a sequence of positive linear operators defined from C2π(R) into C2π(R). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to define some new types of summability methods for double sequences involving the ideas of de la Vall'ee-Poussin mean in the framework of probabilistic normed spaces and establish some interesting results. [ABSTRACT FROM AUTHOR]

A new summabilitymethod of series is introduced and studied. The particular cases of thismethod are, for example, variable-order Cesaro and Riesz methods. Applications to divergence problem of Fourier series are given. An extension of Kolmogorov, Schipp, and Bočkarev's well-known theorems on divergence of Fourier trigonometric,Walsh, and orthonormal series is established. [ABSTRACT FROM AUTHOR]

Various investigators such as Khan (1974), Chandra (2002), and Liendler (2005) have determined the degree of approximation of 2Π-periodic signals (functions) belonging to Lip(α, r) class of functions through trigonometric Fourier approximation using different summability matrices with monotone rows. Recently, Mittal et al. (2007 and 2011) have obtained the degree of approximation of signals belonging to Lip(α, r)- class by general summability matrix, which generalize some of the results of Chandra (2002) and results of Leindler (2005), respectively. In this paper, we determine the degree of approximation of functions belonging to Lip α and W(Lr , ξ(t)) classes by using Cesáro-Nörlund (C1 · Np) summability without monotonicity condition on {pn}, which in turn generalizes the results of Lal (2009).We also note some errors appearing in the paper of Lal (2009) and rectify them in the light of observations of Rhoades et al. (2011). [ABSTRACT FROM AUTHOR]

We study the completeness property and the basis property of the root function system of the Sturm-Liouville operator defined on the segment [0, 1]. All possible types of two-point boundary conditions are considered. [ABSTRACT FROM AUTHOR]

A known theorem, Nigam (2010) dealing with the degree of approximation of conjugate of a signal belonging to Lipξ(t)-class by (E, 1)(C, 1) product summability means of conjugate series of Fourier series has been generalized for the weighted W(Lr,ξ(t)), (r ≥ 1),(t > 0)-class, where ξ(t) is nonnegative and increasing function of t, by En¹Cn¹ which is in more general form of Theorem 2 of Nigam and Sharma (2011). [ABSTRACT FROM AUTHOR]

For the first time, a theorem on double matrix summability of double conjugate Fourier series is established. [ABSTRACT FROM AUTHOR]

Recently, we have proved a main theorem dealing with the absolute Norlund summability factors of infinite series by using δ-quasimonotone sequences. In this paper, we prove that result under weaker conditions. A new result has also been obtained. [ABSTRACT FROM AUTHOR]

By adopting some new ideas, we obtain the estimates of an absolute convergence for the functions of the bounded variation in two variables. Our results generalize the related results of Humphreys and Bojanic (1999) and Wang and Yu (2003) from one dimension to two dimensions and can be applied to several summability methods. [ABSTRACT FROM AUTHOR]

In the work of Bor (2008), we have proved a result dealing with |N̄, pn, θn|ksummability factors by using a quasi-ß-power increasing sequence. In this paper, we prove that result under less and more weaker conditions. Some new results have also been obtained. [ABSTRACT FROM AUTHOR]

By considering Melham's sums (Melham, 2004), we compute various more general nonalternating sums, alternating sums, and sums that alternate according to (-1) (n+1/2) involving the generalized Fibonacci and Lucas numbers. [ABSTRACT FROM AUTHOR]

Let F be a distribution in D' and let f be a locally summable function. The composition F(f(x)) of F and f is said to exist and be equal to the distribution h(x) if the limit of the sequence {Fn(f(x))} is equal to h(x), where Fn(x) = F(x) * δn(x) for n = 1, 2, … and {δn(x)} is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition δ(rs-1)((tanhx+)1/r) exists and Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. for r, s = 1, 2, …, where Kk is the integer part of (s - k - 1)/2 and the constants cj,k are defined by the expansion Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed., for k = 0, 1, 2, …. Further results are also proved. [ABSTRACT FROM AUTHOR]

Let F be a distribution in D' and let f be a locally summable function. The composition F(f(x)) of F and f is said to exist and be equal to the distribution h(x) if the limit of the sequence {Fn(f(x))} is equal to h(x), where Fn(x) = F(x) * δn(x) for n = 1, 2, … and {δn(x)} is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition δ(s)[(sinh-1x+)r] does not exists. In this study, it is proved that the neutrix composition δ(s)[(sinh-1x+)r] exists and is given by Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed., for s = 0, 1, 2, … and r = 1, 2, …, where cs,k,i = (-1)ss![(k - 2i + 1)rs-1 + (k - 2i - 1)rs+r-1]/(2(rs + r - 1)!). Further results are also proved. [ABSTRACT FROM AUTHOR]

The dilogarithm integral Li(xs) and its associated functions Li+(xs) and Li-(xs) are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of these functions and other functions are then found. [ABSTRACT FROM AUTHOR]

A new theorem on (N, p, q)(E, 1) summability of Fourier series has been established. [ABSTRACT FROM AUTHOR]

We give new proof of a four-variable reciprocity theorem using Heine's transformation, Watson's transformation, and Ramanujan's 1ψ1-summation formula. We also obtain a generalization of Jacobi's triple product identity. [ABSTRACT FROM AUTHOR]

We prove that the ratio of the Newman sum over numbers multiple of a fixed integer, which is not a multiple of 3, and the Newman sum over numbers multiple of a fixed integer divisible by 3 is o(1) when the upper limit of summing tends to infinity. We also discuss a connection of our results with a digit conjecture on primes. [ABSTRACT FROM AUTHOR]

Some formulas relating the classical sums of reciprocal powers are derived in a compact way by using generating functions. These relations can be conveniently written by means of certain numbers which satisfy simple summation formulas. The properties of the generating functions can be further used to easily calculate several series involving the classical sums of reciprocal powers. [ABSTRACT FROM AUTHOR]

New results concerning product summability of an infinite series are given. Some special cases are also deduced. [ABSTRACT FROM AUTHOR]