Statistical approximation of certain positive linear operators constructed by means of the Chan–Chyan–Srivastava polynomials

doi:10.1016/j.amc.2006.01.090

Applied Mathematics and Computation

Volume 182, Issue 1, 1 November 2006, Pages 213-222

https://doi.org/10.1016/j.amc.2006.01.090 Get rights and content

Abstract

In this study, by obtaining some Korovkin type approximation results in statistical sense for certain positive linear operators constructed by means of the Chan–Chyan–Srivastava multivariable polynomials [W.-C.C. Chan, C.-J. Chyan, H.M. Srivastava, The Lagrange polynomials in several variables, Integral Transform. Spec. Funct. 12 (2001) 139–148], we show that our approximation method is stronger than the corresponding classical aspects in the approximation theory settings. Furthermore, we investigate their statistical rates by means of the modulus of continuity and the elements of the Lipschitz class.

Introduction

Matrix summability methods of Cesàro type are quite useful in approximation theory, especially in the study of various sequences of linear operators such as the interpolation operator of the Hermite–Fejér class [1]. The Cesàro summability method also corrects the Gibbs phenomenon of some non-positive approximation operators such as the partial sums of Fourier series [12]. Recently, another form of regular (non-matrix) summability transformations, the so-called A-statistical convergence, has been shown to be quite effective in summing non-convergent sequences of positive linear operators (see, for instance, [4], [6], [10]).
Let $A ≔ (a_{jn}) (j, n \in N ≔ {1, 2, 3, \dots})$ be a non-negative regular summability matrix. Recall that the regularity conditions on a matrix A are known as Silverman–Toeplitz conditions in the functional analysis (see, for details, [2]). Then a sequence x ≔ {x_n} is said to be A-statistically convergent to a number L if, for every ε > 0, $\lim_{j \to \infty} \sum_{n : | x_{n} - L | ≧ ε} a_{jn} = 0,$ which is denoted (for convenience) by (cf. [8]) ${stat}_{A} \lim_{n} x_{n} = L .$ Upon replacing the matrix A by the identity matrix I, the A-statistical convergence reduces to the ordinary convergence. Here we should remark that every convergent sequence is A-statistically convergent to the same value for any non-negative regular summability matrix A, but the converse is not true. Especially, Kolk [11] proved that the A-statistical convergence is stronger than the ordinary convergence whenever the non-negative regular matrix A = (a_jn) satisfies the following property: $\lim_{j} \max_{n} {a_{jn}} = 0 .$ Furthermore, by choosing A = C₁, the Cesàro matrix of order one, the A-statistical convergence coincides with the statistical convergence (see, for instance, [7], [9]).

In this paper we use the concept of the A-statistical convergence in the multivariate approximation theory. We first introduce a sequence of positive linear operators which are constructed by means of the Chan–Chyan–Srivastava multivariable polynomials (see, for details, [3]). Then, by obtaining some Korovkin type approximation results in statistical sense for these positive linear operators, we show that our approximation method is stronger than the classical aspects in the approximation theory settings. We also investigate their statistical rates by means of the modulus of continuity and the elements of the Lipschitz class.

Section snippets

Construction of a family of positive linear operators

The familiar (two-variable) polynomials $g_{m}^{(α, β)} (u, v)$ generated by $(1 - ut)^{- α} (1 - vt)^{- β} = \sum_{m = 0}^{\infty} g_{m}^{(α, β)} (u, v) t^{m} (| t | < \min {| u |^{- 1}, | v |^{- 1}})$ are known as the Lagrange polynomials which occur in certain problems in statistics [5, p. 267] (see also [13, p. 441 et seq.]). The multivariable extension of (2.1) generated by $\prod_{j = 1}^{r} {(1 - {tu}_{j})^{- α_{j}}} = \sum_{m = 0}^{\infty} g_{m}^{(α_{1}, \dots, α_{r})} (u_{1}, \dots, u_{r}) t^{m} (| t | < \min {| u_{1} |^{- 1}, \dots, | u_{r} |^{- 1}})$ was introduced and investigated systematically by Chan et al. [3]. In this r-variable case, (2.2) yields the following explicit

Properties of the A-statistical approximation

In this section, we investigate some approximation properties of the positive linear operators $L_{n}^{u^{(1)}, \dots, u^{(r)}} (f; x)$ given by (2.6) via the concept of A-statistical convergence.

We start by considering the case r = 2 in the definition (2.6). In this case, we find from (2.6) that $L_{n}^{u^{(1)}, u^{(2)}} (f; x) ≔ {(1 - {xu}_{n}^{(1)})}^{n} {(1 - {xu}_{n}^{(2)})}^{n} \cdot \sum_{m = 0}^{\infty} \{\sum_{k_{1} + k_{2} = m} f (\frac{k_{2}}{n + k_{2} - 1}) \frac{{(u_{n}^{(1)})}^{k_{1}}}{k_{1}!} \frac{{(u_{n}^{(2)})}^{k_{2}}}{k_{2}!} (n)_{k_{1}} (n)_{k_{2}}\} x^{m} (f \in C [0, 1]; x \in [0, 1]; 0 < u_{n}^{(1)}, u_{n}^{(2)} < 1; n \in N) .$ Then we derive the following preliminary results.

Lemma 1

For each x ∈ [0, 1] and $n \in N$ , $L_{n}^{u^{(1)}, u^{(2)}} (f_{0}; x) = f_{0} (x) =$

Rates of A-statistical convergence

In this section, we compute the rates of A-statistical convergence of our sequence of positive linear operators in Theorem 2 by means of the modulus of continuity and the elements of Lipschitz class.

Let f ∈ C[0, 1]. The modulus of continuity of f denoted by w(f,δ) is defined by $w (f, δ) ≔ \sup_{| y - x | < δ; x, y \in [0, 1]} | f (y) - f (x) | .$

The modulus of continuity of the function f in C[0, 1] provides the maximum oscillation of f in any interval of length not exceeding δ > 0. It is well known that a necessary and sufficient

Acknowledgements

The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP007353.

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Statistical approximation of certain positive linear operators constructed by means of the Chan–Chyan–Srivastava polynomials

Abstract

Introduction

Section snippets

Construction of a family of positive linear operators

Properties of the A-statistical approximation

Rates of A-statistical convergence

Acknowledgements

Summability of Hermite–Fejér interpolation for functions of bounded variation

J. Natur. Sci. Math.

Classical and Modern Methods in Summability

The Lagrange polynomials in several variables

Integral Transform. Spec. Funct.

A-statistical convergence of approximating operators

Math. Inequal. Appl.