Matrix characterization of A-statistical convergence of double sequences

We characterize the multiplier space of summability fields of four dimensional RH-regular matrices and show that the space of multipliers of a nonnegative RH-regular matrix over an algebra \(\mathcal{U} \) is the space of A-statistically convergent double sequences. For this purpose we prove a variant of the Brudno–Mazur–Orlicz bounded consistency theorem for a class of four dimensional matrices. Finally we give a matrix characterization of A-statistical convergence over the space of the Pringsheim A-uniformly integrable double sequences.     

Inclusion results on statistical cluster points

We study the concepts of statistical cluster points and statistical core of a sequence for A _λ methods defined by deleting some rows from a nonnegative regular matrix A. We also relate A _λ-statistical convergence to A _μ-statistical convergence. Finally we give a consistency theorem for A-statistical convergence and deduce a core equality result.     

A Baskakov type generalization of statistical Korovkin theory

In this paper using the notion of A-statistical convergence, where A is a nonnegative regular summability matrix, we obtain some statistical variants of Baskakov's results on the Korovkin type approximation theorems.     

λ-STATISTICAL CONVERGENCE OF ORDER α

In this paper,we introduce the concept of λ-statistical convergence of order α.Also some relations between the λ-statistical convergence of order α and strong(V,λ)-summability of order α are given.     

Statistical approximation properties of high order operators constructed with the Chan-Chyan-Srivastava polynomials

In this paper, by including high order derivatives of functions being approximated, we introduce a general family of the linear positive operators constructed by means of the Chan-Chyan-Srivastava multivariable polynomials and study a Korovkin-type approximation result with the help of the concept of A-statistical convergence, where A is any non-negative regular summability matrix. We obtain a statistical approximation result for our operators, which is more applicable than the classical case. Furthermore, we study the A-statistical rates of our approximation via the classical modulus of continuity.     

Some variations of the Bohman–Korovkin theorem

In this paper we develop the main aspects of the Bohman–Korovkin theorem on approximation of continuous functions with the use of A-statistical convergence and matrix summability method which includes both convergence and almost convergence. Since statistical convergence and almost convergence methods are incompatible we conclude that these methods can be used alternatively to get some approximation results.     

Rates of -statistical convergence of positive linear operators

In this paper we study the rates of A-statistical convergence of sequences of positive linear operators mapping the weighted space C_ρ1 into the weighted space B_ρ2.     

Korovkin-Type Approximation Theorem for Double Sequences of Positive Linear Operators via Statistical A-Summability

In this paper, using the concept of statistical A-summability which is stronger than the A-statistical convergence we provide a Korovkin-type approximation theorem on the space of all continuous real valued functions defined on any compact subset of the real two-dimensional space. We also study the rates of statistical A-summability of positive linear operators.     

On A-invariant mean and A-almost convergence

The idea of A-invariant mean and A-almost convergence is due to J. P. Duran [8], which is a generalization of the usual notion of Banach limit and almost convergence. In this paper, we discuss some important properties of this method and prove that the space F(A) of A-almost convergent sequences is a BK space with ?? · ??_??, and also show that it is a nonseparable closed subspace of the space l _?? of bounded sequences.     

A Voronovskaya-type formula for SMK operators via statistical convergence

In this paper, we obtain a statistical Voronovskaya-type theorem for the Szász-Mirakjan-Kantorovich (SMK) operators by using the notion of A-statistical convergence, where A is a non-negative regular summability matrix.     

Strong and A-statistical comparisons for sequences

Let T and A be two nonnegative regular summability matrices and W(T,p)∩l^∞ and c_A(b) denote the spaces of all bounded strongly T-summable sequences with index p>0, and bounded summability domain of A, respectively. In this paper we show, among other things, that is a multiplier from W(T,p)∩l^∞ into c_A(b) if and only if any subset K of positive integers that has T-density zero implies that K has A-density zero. These results are used to characterize the A-statistical comparisons for both bounded as well as arbitrary sequences. Using the concept of A-statistical Tauberian rate, we also show that is not a multiplier from W(T,p)∩l^∞ into c_A(b) that leads to a Steinhaus type result.     

An algebraic characterization of closed small attainability subspaces of delay systems

If small attainability subspaces of linear time delay systems are closed in a certain Sobolev space, the existence of Lagrange multipliers for optimal control to small solutions is guaranteed. This paper characterizes the required closedness property using an algebraic approach due to B. Jakubczyk. As a main result it turns out that closedness is—in an algebraic sense—generic in the variety of system matrices (A₀,A₁, B₀) with rank A₁ not greater than the dimension of the control space. This is in contrast to known results on closedness of attainability subspaces playing an analogous role for optimal control to fixed final states instead of small solutions.     

A-Statistical extension of the Korovkin type approximation theorem

In this paper, using the concept ofA-statistical convergence which is a regular (non-matrix) summability method, we obtain a general Korovkin type approximation theorem which concerns the problem of approximating a functionf by means of a sequenceL _n f of positive linear operators.     

Rates of Ideal Convergence for Approximation Operators

In this paper, we study a general Korovkin-type approximation theory by using the notion of ideal convergence which includes many convergence methods, such as, the usual convergence, statistical convergence, A-statistical convergence, etc. We mainly compute the rate of ideal convergence of sequences of positive linear operators.     

Korovkin-type Theorems for Abstract Modular Convergence

We give some Korovkin-type theorems on convergence and estimates of rates of approximations of nets of functions, satisfying suitable axioms, whose particular cases are filter/ideal convergence, almost convergence and triangular A-statistical convergence, where A is a non-negative summability method. Furthermore, we give some applications to Mellin-type convolution and bivariate Kantorovich-type discrete operators.     

On some ‘duality’ of the Nikodym property and the Hahn property

Drewnowski and Paúl proved in [L. Drewnowski, P.J. Paúl, The Nikodým property for ideals of sets defined by matrix summability methods, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94 (2000) 485-503] that for any strongly nonatomic submeasure η on the power set P(N) of N the ideal Z(η)={N∈P(N)|η(N)=0} has the Nikodym property (NP); in particular, this result applies to densities dA defined by strongly regular matrices A. Grahame Bennett and the authors stated in [G. Bennett, J. Boos, T. Leiger, Sequences of 0's and 1's, Studia Math. 149 (2002) 75-99] that the strong null domain ₀|A| of any strongly regular matrix A has the Hahn property (HP). Moreover, Stuart and Abraham [C.E. Stuart, P. Abraham, Generalizations of the Nikodym boundedness and Vitali-Hahn-Saks theorems, J. Math. Anal. Appl. 300 (2) (2004) 351-361] pointed out that the said results are in some sense dual and that the last one follows from the first one by considering the density dA (defined by A) as submeasure on P(N) and the ideal Z(dA) as well by identifying P(N) with the set χ of sequences of 0's and 1's. In this paper we aim at a better understanding of the intimated duality and at a characterization of those members of special classes of matrices A such that Z(dA) has NP (equivalently, ₀|A| has HP).     

Two valued measure and summability of double sequences in asymmetric context

We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called ??-statistical convergence/Cauchy condition and convergence/Cauchy condition in ??-density, studied for real numbers in our recent paper [7]) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz [15] and Mursaleen and Edely [17]. We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case [7] but at the same time, like [3], shows that symmetry is not essential for any result of [7] and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).     

I-convergence theorems for a class of k-positive linear operators

In this paper, we obtain some approximation theorems for k- positive linear operators defined on the space of analytical functions on the unit disc, via I-convergence. Some concluding remarks which includes A-statistical convergence are also given.      

Improved gradient method for monotone and lipschitz continuous mappings in banach spaces

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {A_n}_(n∈N) be a family of monotone and Lipschitz continuos mappings of C into E~*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for{A_n} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.     

On lacunary statistical convergence of order α

In this article, we introduce the concept of lacunary statistical convergence of order α of real number sequences and give some inclusion relations between the sets of lacunary statistical convergence of order α and strong N_θ^α(p)-summability. Furthermore, some relations between the spaces N_θ^αS_θ^α are examined.