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KYUNGPOOK Math. J. 2019; 59(4): 679-688

Published online December 23, 2019

Copyright © Kyungpook Mathematical Journal.

Statistical Approximation of Szász Type Operators Based on Charlier Polynomials

Arun Kajla

Department of Mathematics, Central University of Haryana, Haryana-123031, India
e-mail : rachitkajla47@gmail.com

Received: January 13, 2018; Revised: December 13, 2018; Accepted: December 27, 2018

Abstract

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In the present note, we study some approximation properties of the Szász type operators based on Charlier polynomials introduced by S. Varma and F. Taşdelen (Math. Comput. Modelling, 56 (5–6) (2012) 108–112). We establish the rates of A-statistical convergence of these operators. Finally, we prove a Voronovskaja type approximation theorem and local approximation theorem via the concept of A-statistical convergence.

Keywords: Szá,sz operator, Charlier polynomials, modulus of continuity, statistical convergence, bounded variation.

1. Introduction

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Szász [28] defined the following linear positive operators

Sn(f;x)=e-nx∑k=0∞(nx)kk!f(kn),

where x ∈ [0,∞) and f(x) is a continuous function on [0,∞) whenever the above sum converges uniformly. Many authors have studied approximation properties of these operators and generalized Szász operators involving different type of polynomials. Jakimovski and Leviatan [19] introduced a generalization of Szász operators based on the Appell polynomials and studied the approximation properties of these operators. Varma et al. [29] proposed the generalization of Szász operators based on Brenke type polynomials and gave convergence properties by means of a Korovkin type theorem and an order approximation using classical methods. Aral and Duman [4] studied a statistical asymptotic formula for the Szász-Mirkajan-Kantorovich operators via A–stastical convergence. In 2011, Örkcü and Doǧru [25] defined a Kantorovich variant of q–Szász-Mirakjan operators and obtained weighted statistical convergence theorems for the operators. Özarslan [26] proposed the Mittag-Leffler operators and obtained some direct theorems and A–statistical approximation theorems for these operators. Altomare et al. [1] considered a new kind of generalization of SMK operators and studied the rate of approximation by interms suitable moduli of continuity. Several researchers have studied various generalizations of these operators and obtained results about their approximation properties; we refer the reader to such papers as [3, 5, 6, 11, 13, 14, 15, 16, 17, 20, 22, 23, 24].

Varma and Taşdelen introduced a relation between orthogonal polynomials and linear positive operators. They considered Szász type operators depending on Charlier polynomials.

These polynomials [18] have generating functions of the form

et(1-ta)u=∑k=0∞Ck(a)(u)tkk!,   ∣t∣<a,

where Ck(a)(u)=∑r=0k(kr) (-u)r(1a)r and (m)0 = 1, (m)j = m(m+1)···(m+j−1) for j ≥ 1.

For Cγ[0,∞) := {f ∈ C[0,∞) : |f(t)| ≤ Meγt for some γ > 0,M > 0 and t ∈ [0,∞)}, Varma and Taşdelen [30] studied the following Szász type operators based on Charlier polynomials

Ln(f;x,a)=e-1(1-1a)(a-1)nx∑k=0∞Ck(a)(-(a-1)nx)k!f(kn),

where a > 1 and x ∈ [0,∞). If a → ∞ and x-1n instead of x, these operators reduce to the operators (1.1). They studied uniform convergence of these operators by using Korovkin’s theorem on compact subsets of [0,∞) and the order of approximation by using the modulus of continuity. Very recently, Kajla and Agrawal [21] discussed some approximation properties of these operators e.g. Lipschitz type space, weighted approximation theorems and rate of approximation of functions having derivatives of bounded variation.

The aim of the present article is to study A–statistical convergence to prove a Korovkin type convergence theorem. We also obtain a Voronovskaja type approximation theorem and local approximation theorem via A–statistical convergence.

Lemma 1.1.([21])

For the operators ℒn(ts; x, a), s = 3, 4, we have

(i) Ln(t3;x,a)=x3+x2n(6+3a-1)+2xn2(1(a-1)2+3a-1+5)+5n3;

(ii) Ln(t4;x,a)=x4+x3n(10+6a-1)+x2n2(31+30a-1+11(a-1)2)+xn3(67+31a-1+20(a-1)2+6(a-1)3)+15n4.

Lemma 1.2.([21])

For the operators ℒn(f; x, a), we have

(i) Ln((t-x);x,a)=1n;

(ii) Ln((t-x)2;x,a)=axn(a-1)+2n2;

(iii) Ln((t-x)4;x,a)=xn3(17+49(a-1)-20(a-1)2+6(a-1)3)+x2n2(19-46(a-1)+3(a-1)2)+15n4.

In what follows, let C̃B[0,∞) be the space of all real valued bounded and uniformly continuous functions f on [0,∞), endowed with the norm ‖f‖C˜B[0,∞]=supx∈[0,∞)∣f(x)∣.

Further, let us define the following Peetre’s K-functional:

K2(f,δ)=infg∈W2}‖f-g‖+δ‖g″‖},δ>0,

where W2 = {g ∈ C̃B[0,∞) : g′, g″ ∈ C̃B[0,∞)}. By [7, p.177, Theorem 2.4], there exists an absolute constant M >0 such that

K2(f,δ)≤M{ω2(f,δ)+min(1,δ)   ‖f‖C˜B[0,∞)},

where the second order modulus of smoothness is defined as

ω2(f,δ)=sup0<∣h∣≤δsupx∈[0,∞)∣f(x+2h)-2f(x+h)+f(x)∣.

We define the usual modulus of continuity of f ∈ C̃B[0,∞) as

ω(f,δ)=sup0<∣h∣≤δsupx∈[0,∞)∣f(x+h)-f(x)∣.

Let Bϕ[0,∞) be the space of all real valued functions on [0,∞) satisfying the condition |f(x)| ≤ Mfϕ(x), where Mf is a positive constant depending only on f and ϕ(x) = 1 + x2 is a weight function. Let Cϕ[0,∞) be the space of all continuous functions in Bϕ[0,∞) with the norm ‖f‖ϕ:=supx∈[0,∞)∣f(x)∣ϕ(x) and Cϕ*[0,∞):={f∈Cϕ[0,∞):limx→∞∣f(x)∣ϕ(x)   is finite}.

2. Main Results

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2.1. A-statistical Convergence

First, we give some basic definitions and notations on the concept of A-statistical convergence. Let A = (ank), (n, k ∈ ℕ), be a non-negative infinite summability matrix. For a given sequence x := (xk), the A-transform of x denoted by Ax : ((Ax)n) is defined as

(Ax)n=∑k=1∞ankxk,

provided the series converges for each n. A is said to be regular if limn(Ax)n=L whenever limnxn=L. The sequence x = (xn) is said to be a A-statistically convergent to L i.e. stA-limnxn=L if for every ε > 0, limn∑k:∣xk-L∣≥ɛank=0.

Replacing A by C1, the Cesáro matrix of order one, the A-statistical convergence reduces to the statistical convergence. Similarly, if we take A = I, the identity matrix, then A-statistical convergence coincides with the ordinary convergence. Many researchers have studied the statistical convergence of different types of operators (cf. [2, 8, 9, 10, 12, 25, 27]). In the following result we prove a weighted Korovkin theorem via A-statistical convergence.

Theorem 2.1

Let A = (ank) be a non-negative regular summability matrix and x ∈ [0,∞). Then, for allf∈Cϕ*[0,∞)we have

stA-limn‖Ln(f,.,a)-f‖ϕα=0,

where ϕα(x) = 1+x2+α, α>0.

Proof

From [10, p. 191, Th. 3], it is sufficient to show that stA-limn‖Ln(ej,.,a)-ej‖ϕ=0, where ej(x) = xj, j = 0, 1, 2.

In view of [30, Lemma 1], it follows that

stA-limn‖Ln(e0,.,a)-e0‖ϕ=0.

Again, using [30, Lemma 1], we find that

‖Ln(e1,.,a)-e1‖ϕ=supx≥0|x+1n-x|(1+x2)≤1n.

For ε > 0, let us define the following sets:

ℱ:={n∈ℕ:‖Ln(e1;.,a)-e1‖ϕ≥ɛ}

and

ℱ1:={n∈ℕ:1n≥ɛ}.

Then, we obtain ℱ ⊆ ℱ1 which implies that ∑k∈ℱank≤∑k∈ℱ1ank and hence

stA-limn‖Ln(e1;.,a)-e1‖ϕ=0.

Next, we can write

‖Ln(e2;.,a)-e2‖ϕ=supx≥011+x2|x2+xn(3+1a-1)+2n2-x2|≤1n(3a-2)(a-1)+2n2.

Now, we define the following sets:

G:={n∈ℕ:‖Ln(e2;.,a)-e2‖ϕ≥ɛ},G1:={n∈ℕ:1n(3a-2)(a-1)≥ɛ2}andG2:={n∈ℕ:2n2≥ɛ2}.

Then, we get , which implies that

∑k∈Gank≤∑k∈G1ank+∑k∈G2ank

and hence

stA-limn‖Ln(e2;.,a)-e2‖ϕ=0.

This completes the proof of the theorem.

Similarly, from Lemma 1.2, we have

stA-limn‖Ln((e1-xe0)j;x,a‖ϕ=0,   j=0,1,2,3,4.

Next, we prove a Voronovskaja type theorem for the operators ℒn.

Theorem 2.2

Let A = (ank) be a nonnegative regular summability matrix. Then, for everyf∈Cϕ*[0,∞)such that f′, f″∈Cϕ*[0,∞), we have

stA-limn→∞n(Ln(f;x,a)-f(x))=f′(x)+12ax(a-1)f″(x),

uniformly with respect to x ∈ [0,E], (E > 0).

Proof

Let f, f′, f″∈Cϕ*[0,∞). For each x ≥ 0, define a function

Θ(t,x)={f(t)-f(x)-(t-x)f′(x)-12(t-x)2f″(x)(t-x2)if t≠x0,if t=x.

Then

Θ(x,x)=0   and   Θ(·,x)∈Cϕ*[0,∞).

Thus, we have

f(t)=f(x)+(t-x)f′(x)+12(t-x)2f″(x)+(t-x)2Θ(t,x).

Operating by ℒn on the above equality, we obtain

n(Ln(f;x,a)-f(x))=f′(x)nLn((t-x);x,a)+12f″(x)nLn((t-x2);x,a)+nLn((t-x)2Θ(t,x);x,a).

In view of Lemma 1.2, we get

stA-limn→∞nLn((t-x);x,a)=1,stA-limn→∞nLn((t-x)2;x,a)=ax(a-1),

and

stA-limn→∞n2Ln((t-x)4;x,a)=x2(19-46(a-1)+3(a-1)2),

uniformly with respect to x ∈ [0,E].

Applying Cauchy-Schwarz inequality, we have

nLn((t-x)2Θ(t,x);x,a)≤n2Ln((t-x)4;x,a)Ln(Θ2(t,x);x,a).

Let η(t, x) = Θ2(t, x), we observe that η(x, x) = 0 and η(·,x)∈Cϕ*[0,∞). It follows from the proof of Theorem 2.1 that

stA-limn→∞Ln(Θ2(t,x);x,a)=stA-limn→∞Ln(η(t,x);x,a)=η(x,x)=0,

uniformly with respect to x ∈ [0,E]. Now, using (2.7), we obtain

stA-limn→∞nLn((t-x)2Θ(t,x);x,a)=0.

Combining (2.5), (2.6) and (2.8), we get the desired result.

Now, we obtain the rate of A-statistical convergence for the operators ℒn with the help of Peetre’s K-functional.

Theorem 2.3

Let f ∈ W2. Then, we have

stA-limn‖Ln(f;x,a)-f(x)‖C˜B[0,∞)=0.
Proof

By our hypothesis, from Taylor’s expansion we find that

Ln(f;x,a)-f(x)=f′(x)Ln((e1-x);x,a)+12f″(χ)Ln((e1-x)2;x,a);

where χ lies between t and x.

Thus, we get

‖Ln(f;x,a)-f(x)‖C˜B[0,∞)≤‖f′‖C˜B[0,∞)‖Ln((e1-x);x,a)‖C[0,b]+‖f″‖C˜B[0,∞)‖Ln((e1-x)2;x,a)‖C[0,b]

Using (2.4) for ε > 0, we have

limn∑k∈ℕ:‖f′‖C˜B[0,∞)‖Lk((e1-x);x,a)‖C[0,b]≥ɛ2ank=0,limn∑k∈ℕ:‖f″‖C˜B[0,∞)‖Lk((e1-x)2;x,a)‖C[0,b]≥ɛ2ank=0.

From (2.10), we may write

∑k∈ℕ:‖Lk(f;x,a)-f(x)‖C˜B[0,∞)≥ɛank≤∑k∈ℕ:‖f′‖C˜B[0,∞)‖Lk((e1-x);x,a)‖C[0,b]≥ɛ2ank+∑k∈ℕ:‖f″‖C˜B[0,∞)‖Lk((e1-x)2;x,a)‖C[0,b]≥ɛ2ank.

Hence taking limit as n→∞, we get the desired result.

Theorem 2.4

Let f ∈ C̃B[0,∞), we have

‖Ln(f;x,a)-f(x)‖C˜B[0,∞)≤Mω2(f,δna),

wherδna=‖Ln((e1-x);x,a)‖C˜B[0,∞]+‖Ln((e1-x)2;x,a)‖C˜B[0,∞]e.

Proof

Let g ∈ W2, by (2.10), we have

‖Ln(g;x,a)-g(x)‖C˜B[0,∞)≤‖Ln((e1-x);x,a)‖C˜B[0,∞)‖g′‖C˜B[0,∞)+12‖Ln((e1-x)2;x,a)‖C˜B[0,∞)‖g″‖C˜B[0,∞)≤δna‖g‖W2.

For every f ∈ C̃B[0,∞) and g ∈ W2, we get

‖Ln(f;x,a)-f(x)‖C˜B[0,∞)≤‖Ln(f;x,a)-Ln(g;x,a)‖C˜B[0,∞)+‖Ln(g;x,a)-g(x)‖C˜B[0,∞)+‖g-f‖C˜B[0,∞)≤2‖g-f‖C˜B[0,∞)+‖Ln(g;x,a)-g(x)‖C˜B[0,∞)≤2‖g-f‖C˜B[0,∞)+δna‖g‖W2.

Taking the infimum on the right hand side over all g ∈ W2, we obtain

‖Ln(f;x,a)-f(x)‖C˜B[0,∞)≤2K2(f,δna).

Using (1.4), we have

‖Ln(f;x,a)-f(x)‖C˜B[0,∞)≤M{ω2(f,δna)+min(1,δna)‖f‖C˜B[0,∞)}

From (2.4), we get stA-limnδna=0, hence stA-ω2(f,δna)=0. Therefore we get the rate of A-statistical convergence of the sequence of the operators ℒn(f; x, a) defined by (1.3) to f(x) in the space C̃B[0,∞). If we take A = I, we obtain the ordinary rate of convergence of these operators.

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