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Journal of Applied Analysis

Editor-in-Chief: Liczberski, Piotr / Ciesielski, Krzysztof

Managing Editor: Gajek, Leslaw

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Volume 24, Issue 1

Issues

Deferred weighted 𝒜-statistical convergence based upon the (p,q)-Lagrange polynomials and its applications to approximation theorems

H. M. Srivastava
  • Corresponding author
  • Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada; and Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, P. R. China
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Bidu Bhusan Jena / Susanta Kumar Paikray / U. K. Misra
  • Department of Mathematics, National Institute of Science and Technology, Palur Hills, Golanthara 761008, Odisha, India
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Published Online: 2018-05-03 | DOI: https://doi.org/10.1515/jaa-2018-0001

Abstract

Recently, the notion of positive linear operators by means of basic (or q-) Lagrange polynomials and 𝒜-statistical convergence was introduced and studied in [M. Mursaleen, A. Khan, H. M. Srivastava and K. S. Nisar, Operators constructed by means of q-Lagrange polynomials and A-statistical approximation, Appl. Math. Comput. 219 2013, 12, 6911–6918]. In our present investigation, we introduce a certain deferred weighted 𝒜-statistical convergence in order to establish some Korovkin-type approximation theorems associated with the functions 1, t and t2 defined on a Banach space C[0,1] for a sequence of (presumably new) positive linear operators based upon (p,q)-Lagrange polynomials. Furthermore, we investigate the deferred weighted 𝒜-statistical rates for the same set of functions with the help of the modulus of continuity and the elements of the Lipschitz class. We also consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper.

Keywords: Statistical convergence; Korovkin-type approximation theorems; rate of convergence; modulus of continuity; Lipschitz class; Lagrange polynomials; Chan–Chyan–Srivastava polynomials

MSC 2010: 40A05; 41A36; 40G15

1 Introduction, definitions and motivation

In the study of sequence spaces, classical convergence has got innumerable applications where the convergence of a sequence requires that almost all elements are to satisfy the convergence condition, that is, all the elements of the sequence need to be in an arbitrarily small neighborhood of the limit. However, such restriction is relaxed in statistical convergence, where the validity of the convergence condition is achieved only for a majority of elements. The notion of statistical convergence was introduced and studied by Fast [12] and Steinhaus [36]. Recently, statistical convergence has been a dynamic research area, basically due to the fact that it is more general than classical convergence, and such theory is discussed in the study in the areas of (for instance) Fourier Analysis, Number Theory and Approximation Theory. Gradually the idea has been extended to different summability methods. For more details of statistical convergence as well as statistical summability, see the recent works [6, 14, 16, 19, 29, 31, 32, 33, 35].

An application of statistical summability gave rise to the theory of statistical approximation, which has been an active area of research in the last decade. For more details, the statistical approximation properties have been investigated for q-analogues of several operators. For example, in [2] a q-analogue of Butzer and Hahn operators, in [3, 24] a q-analogue of Stancu–Beta operators, in [11] a q-Bleimann–Butzer–Hahn operator, in [15] a q-Baskakov–Kantorovich operator, in [18] a q-analogue of the Meyer–König and Zeller operator, in [34] a q-Szász–Mirakjan–Kantorovich-type operator, in [28] a q-Szász–Mirakjan operator, and in [30] a q-analogue of the Bernstein–Kantorovich operator were defined and their statistical approximation properties were investigated.

Subsequently, the statistical approximation properties have also been investigated for (p,q)-analogues of several operators. For example, Mursaleen et al. [21] first applied the concept of the (p,q)-calculus in approximation theory as an alternative to the q-calculus. Later, based upon (p,q)-integers, a (p,q)-analogue of the Bernstein–Stancu operators [22], a Kantorovich variant of the (p,q)-Szász–Mirakjan operators [20], the (p,q)-Lorentz polynomials on a compact disk [26], a (p,q)-analogue of the Bernstein–Schurer operators [27], and so on, have also been introduced.

Let be the set of natural numbers and let K. Also let

Kn={k:kn and kK}

and suppose that |Kn| is the cardinality of Kn. Then the natural density of K is defined by

d(K)=limn|Kn|n=limn1n|{k:kn and kK}|,

provided that the limit exists.

A given sequence (xn) is said to be statistically convergent to if, for each ϵ>0, the set

Kϵ={k:k and |xk-|ϵ}

has natural density zero (see [12, 36]). This means that, for each ϵ>0, we have

d(Kϵ)=limn|Kϵ|n=limn1n|{k:kn and |xk-|ϵ}|=0.

In this case, we write

statlimnxn=.

We present below an example to illustrate that every convergent sequence is statistically convergent but the converse is not true.

Example 1.1.

Let us consider the sequence x=(xn) by

xn={n,n=m2 for all m,1notherwise.

Then it is easy to see that the sequence (xn) is divergent in the ordinary sense, while 0 is the statistical limit of (xn) since d(K)=0, where

K={m2:m}.

In the year 2009, Karakaya and Chishti [17] introduced the fundamental concept of weighted statistical convergence, and their definition was later modified by Mursaleen, Karakaya, Ertürk and Gürsoy [23].

Suppose that (sk) is a sequence of non-negative numbers with n-th partial sum Sn such that

Sn=k=0nsk(s0>0,n).

Then, by setting

tn=1Snk=0nskxk(n0:={0}),

the given sequence (xn) is said to be weighted statistically convergent (or statN¯-convergent) to a number if, for each ϵ>0, the set

{k:kSn and sk|xk-|ϵ}

has zero weighted density (see [23]). This means that, for each ϵ>0, we have

limn1Sn|{k:kSn and sk|xk-|ϵ}|=0.

In this case, we write

statN¯limxn=.

In the year 2013, Belen and Mohiuddine [4] established a new technique for weighted statistical convergence in terms of the de la Vallée Poussin mean, and it was subsequently investigated further by Braha, Srivastava and Mohiuddine [7] as the Λn-weighted statistical convergence. Very recently, a certain class of weighted statistical convergence and associated Korovkin-type approximation theorems involving trigonometric functions were established by Srivastava, Jena, Paikray and Misra [33].

Let X and Y be two sequence spaces and let 𝒜=(an,k) be an infinite matrix. If, for each xkX, the series given by

𝒜nx=k=1an,kxk

converges for each n and the sequence (𝒜nx) belongs to Y, then we say that the matrix 𝒜 maps X into Y. By the symbol (X,Y) we denote the set of all matrices which map X into Y.

We say that 𝒜 is regular if

limn𝒜nx=wheneverlimkxk=.

The well-known Silverman–Toeplitz theorem (see, for details, [5]) asserts that 𝒜=(an,k) is regular if and only if the following conditions hold:

  • (i)

    supnk=1|an,k|<.

  • (ii)

    limnan,k=0 for each k.

  • (iii)

    limnk=1an,k=1.

Freedman and Sember [13] extended the definition of statistical convergence with the help of the non-negative regular matrix 𝒜=(an,k), which they called the 𝒜-statistical convergence. For any non-negative regular matrix 𝒜, we say that a sequence (xn) is 𝒜-statistically convergent (or stat𝒜-convergent) to a number if, for each ϵ>0, the set

{k:k and |xk-|ϵ}

has zero 𝒜-density. This means that, for each ϵ>0, we have

limnk:|xk-|ϵan,k=0.

In this case, we write

stat𝒜limxn=.

In the year 2016, Mohiuddine [19] introduced the notion of weighted 𝒜-summability by using a weighted regular matrix. He also introduced the definitions of statistically weighted 𝒜-summability and weighted 𝒜-statistical convergence. In particular, he proved a Korovkin-type approximation theorem by means of statistically weighted 𝒜-summable sequences of real or complex numbers. Subsequently, Kadak, Braha and Srivastava [16] have investigated the notion of statistical weighted -summability by using a weighted regular matrix to establish some approximation theorems.

Motivated essentially by the above-mentioned works, here we establish the concept of deferred weighted 𝒜-statistical convergence.

Let (an) and (bn) be sequences of non-negative integers satisfying the following conditions:

  • (i)

    an<bn (n).

  • (ii)

    limnbn=.

The above conditions (i) and (ii) are known as the regularity conditions for the deferred weighted mean (see Agnew [1]).

We next suppose that (sn) is the sequence of non-negative real numbers such that

Sn=m=an+1bnsm.

Now, in order to simply define the deferred weighted mean σn, we first set

σn=1Snm=an+1bnsmxm.

We then say that a sequence (xn) is deferred weighted summable (or cD(N¯)-summable) to if

limnσn=.

In this case, we write

cD(N¯)-limxn=,

and let cD(N¯) denote the space of all deferred weighted summable sequences.

Next, we present below the following definitions.

Definition 1.2.

A sequence (xn) is said to be deferred weighted 𝒜-summable (or [D(N¯)𝒜,pn]-summable) to a number if the 𝒜-transform of (xn) is deferred weighted summable to the same number , that is,

limn1Snm=an+1bnk=1smam,kxk=.

In this case, we write

[D(N¯)𝒜,pn]-xn=.

Here we denote the set of all deferred weighted 𝒜-summable sequences by [D(N¯)𝒜,pn)].

Definition 1.2 is a generalization of several known definitions as discussed in Remark 2.1 below.

Remark 1.3.

If an=0 and bn=n for all n, then the deferred weighted 𝒜-summability reduces to the weighted 𝒜-summability (see [19]). Also, if an=0 and bn=n for all n, and 𝒜=I (I being the identity matrix), then the deferred weighted 𝒜-summability reduces simply to the weighted summability (see [23]). Moreover, if an=0 and bn=n for all n, 𝒜=(𝒞,1) and sn=1, then the deferred weighted 𝒜-summability reduces to the statistical convergence (see [12]).

For simplicity in notation, we shall use the convention that

𝒜nD(N¯)(x)=1Snm=an+1bnk=1smam,kxk.

Definition 1.4.

Let 𝒜=(an,k) be an infinite matrix and let (an) and (bn) be sequences of non-negative integers. The matrix 𝒜=(an,k) is said to be a deferred weighted regular matrix (or deferred weighted regular method) if

𝒜xcD(N¯)for all x=(xn)c,

with

cD(N¯)-limn𝒜x=𝒜-lim(xn).

We denote it by 𝒜(c:cD(N¯)). This means that 𝒜nD(N¯)(x) exists for each n and xc, and

𝒜nD(N¯)(x)(n)  whenever  xn(n).

We denote the set of all such regular matrices (methods) by w+.

As a characterization of the deferred weighted regular methods, we present the following theorem.

Theorem 1.5.

Let A=(an,k) be an infinite matrix. Also let (an) and (bn) be sequences of non-negative integers. Then A(c:cD(N¯)) if and only if

supnk=11Sn|m=an+1bnsmam,k|<,(1.1)limn1Snm=an+1bnsmam,k=0(for each k),(1.2)limn1Snm=an+1bnk=1smam,k=1.(1.3)

Proof.

Assume that (1.1), (1.2) and (1.3) hold true and that xk (k). Then, for each ϵ>0, there exists m0 such that |xk-|ϵ (m>m0). Thus, we have

|𝒜nD(N¯)(x)-|=|1Snm=an+1bnk=1smam,kxk-|=|1Snm=an+1bnk=1smam,k(xk-)+(1Snm=an+1bnk=1smam,k-1)||1Snm=an+1bnk=1smam,k(xk-)|+|||1Snm=an+1bnk=1smam,k-1||1Snm=an+1bnk=1bn-2smam,k(xk-)|+|1Snm=an+1bnk=bn-1smam,k(xk-1)|+|||1Snm=an+1bnk=1smam,k-1|supk|xk-|k=1bn-21Snm=an+1bnsmam,k+ϵ1Snm=an+1bnk=1smam,k+|||1Snm=an+1bnk=1smam,k-1|.

Taking n and using (1.2) and (1.3), we get

|1Snm=an+1bnk=1smam,kxk-|ϵ,

which implies that

limn1Snm=an+1bnk=1smam,kxk==lim(xn)

since ϵ>0 is arbitrary.

Conversely, let 𝒜(c:cD(N¯)) and xkc. Then, since 𝒜x exists, we have the inclusion given by

(c:cD(N¯))(c:).

Clearly, there exists a constant M such that

|1Snm=an+1bnk=1smam,k|M(for all m,n),

and the corresponding series given by

1Snm=an+1bnk=1smam,k

converges absolutely for each n. Therefore, (1.1) is valid.

We now consider the sequence x(n)=(xk(n))c0 defined by

xk(n)={1,n=k,0,nk,

for all n and y=(yk)=(1,1,1,)c. Then, since 𝒜x(n) and 𝒜y belong to cD(N¯), equations (1.2) and (1.3) are fairly obvious. ∎

Next, for the statistical version, we present Definition 1.6 below.

Definition 1.6.

Let 𝒜=(an,k) be a non-negative deferred weighted regular matrix and let K=(ki) (ki<ki+1). Also let (an) and (bn) be sequences of non-negative integers. Then the deferred weighted 𝒜-density of K is defined by

dD(N¯)𝒜(K)=limn1Snm=an+1bnkKsmam,k,

provided that the limit exists.

We say that the sequence (xn) is deferred weighted 𝒜-statistical convergent to the number if, for every ϵ>0, we have

dD(N¯)𝒜(Kϵ)=0,

where

Kϵ={k:k and |xk-|ϵ}.

In this case, we write

stat𝒜D(N¯)limn(xn)=.

2 Construction of a family of positive linear operators

In this section, we construct a sequence of new positive linear operators by means of (p,q)-Lagrange polynomials in order to establish some new approximation results via deferred weighted 𝒜-statistical convergence.

Let us first recall some basic definitions and notations about the (p,q)-integers: For any (n), the (p,q)-integer [n]p,q is defined by

[n]p,q={pn-qnp-q,n1,0,n=0,

where 0<q<p1.

The (p,q)-factorial is defined by

[n]!p,q={[1]p,q[2]p,q[n]p,q,n1,1,n=0.

Also the (p,q)-binomial coefficient is defined by

[nk]p,q=[n]!p,q[k]!p,q[n-k]!p,qfor all n,k with nk.

In the year 2001, Chan, Chyan and Srivastava [8] introduced and studied the following multivariable Lagrange polynomials [8, p. 140, (6)]:

gn(α1,,αr)(x1,,xr)=k1+k2++kr=n(α1)k1(αr)krx1k1k1!xrkrkr!,(2.1)

where

(λ)k=λ(λ+1)(λ+k-1)and(λ)0=1.

By using the above (Chan–Chyan–Srivastava) polynomials, Erkuş, Duman and Srivastava [10, p. 215, (2.6)] introduced the following family of positive linear operators on C[0,1]:

Lnu(1),,u(r)(f;x)=(j=1r(1-xun(j))n)m=0[k1+k2++kr=mf(krn+kr-1)(un(1))k1k1!(un(r))krkr!(n)k1(n)kr]xm,(2.2)

where u(j)=(un(j))n are sequences of real numbers such that

0<un(j)<1(j=0,1,2,,r,n)  and  fC[0,1](x[0,1]).

In the year 2008, Duman [9, p. 540, (2.2)] introduced the q-polynomials gm,q(α1,,αr)(x1,,xr), the q-analogue of Chan–Chyan–Srivastava polynomials in (2.1), which are generated by

(i=1r1(xit;q)αi)=i=1r{(1-txiqk)-αi}=m=0gm,q(α1,,αr)(x1,,xr)tm=m=0[k1+k2++kr=m(qαk1;q)k1(qαk2;q)k2(qαkr;q)krx1k1(q;q)k1xrkr(q;q)kr]tm,

where |t|<min(|x1|,|x2|).

Very recently, Mursaleen, Khan, Srivastava and Nisar [25, p. 6912, (2.4)] investigated the following family of positive linear operators on C[0,1], which happens to be a q-analogue of (2.2):

Ln,qu(1),,u(r)(f;x)=(j=1r(1-xun(j)qr)n)m=0[k1+k2++kr=m(qα1;q)k1(qαr;q)krf((q;q)krn+(q;q)kr-1)(un(1))k1(q;q)k1(un(r))kr(q;q)kr(n;q)k1(n;q)kr]xm,(2.3)

where u(j)=(un(j))n are sequences of real numbers such that

0<un(j)<1(j=0,1,2,,r,n)  and  fC[0,1](x[0,1]).

Here, as before, q (|q|<1) is a real number, the number (λ;q)n is defined by

(λ;q)n=(λ;q)(λqn;q),

and (;) is the infinite version of the so-called q-shifted factorial defined by

(λ;q)=k=0(1-λqk).

As usual, let C[0,1] denote the space of all real-valued continuous functions defined on the closed interval [0,1]. We also consider the usual supremum norm on C[0,1]. Then, by definition, a linear operator L defined on C[0,1] is said to be positive if

L(f)0wheneverf0.

We denote the value of L(f) at a point t[0,1] by L(f(y);t) or, simply, by L(f;t).

We now define the following (presumably new) family of positive linear operators on C[0,1], that is, a (p,q)-analogue of (2.3) given by

Ln,p,qx(1),,x(r)(f;t)=(j=1r(pr-txn(j)qr)n)m=0[k1+k2++kr=m(pα1;qα1)k1(pαr;qαr)krf((p;q)krn+(p;q)kr-1)(xn(1))k1(p;q)k1(xn(r))kr(p;q)kr(p;q)k1(p;q)kr]tm,(2.4)

where

|t|<min(|xn(1)|,|xn(2)|),fC[0,1],t[0,1],0<xn(j)<1(j=0,1,2,,r,n).

Here p and q (0<q<p1) are real numbers, the number (p;q)n is defined by

(p;q)n=(p;q)p,q(pn+1;qn+1)p,q

and (;) is the infinite (p,q)-shifted factorial defined by

(α;β)p,q=k=0(αpk-βqk)(0<β<α1).

Remark 2.1.

If we take p=1, then the (p,q)-Chan–Chyan–Srivastava polynomial reduces to the q-Chan–Chyan–Srivastava polynomial (see [25]).

3 Properties of the deferred weighted 𝒜-statistical convergence

In this section, we investigate some approximation properties of the positive linear operators Ln,p,qx(1),,x(r)(f;t) given by (2.4) with respect to the notion of our newly-defined deferred weighted 𝒜-statistical convergence.

We begin by considering the case when r=2 in equation (2.4), and we have

Ln,p,qx(1),x(2)(f;t)=(j=12(p2-txn(j)q2)n)m=0[k1+k2=m(pα1;qα1)k1(pα2;qα2)k2f((p;q)k2n+(p;q)k2-1)(xn(1))k1(p;q)k1(xn(2))k2(p;q)k2(p;q)k1(p;q)k2]tm,(3.1)

where

|t|<min(|xn(1)|,|xn(2)|),fC[0,1],t[0,1],0<xn(1),xn(2)<1(n).

We now state and prove some preliminary properties in the form of the following lemmas.

Lemma 3.1.

For each t[0,1] and nN,

Ln,p,qx(1),x(2)(f0;t)=1(f0(t)=1).

Proof.

In (2.4), we first set r=2 and α1=α2=n.

We observe that, since

0<xn(1),xn(2)<1(n),

the condition of (2.4) is satisfied for each t[0,1]. Then, by (3.1), we get

Ln,p,qx(1),x(2)(f0;t)=j=12(p2-txn(j)q2)nm=0gm,p,q(n,n)(xn(1),xn(2))tm=1,

where

gm,p,q(n,n)(xn(1),xn(2))tm=k1+k2=m(pn;qn)k1(pn;qn)k2(xn(1))k1(p;q)k1(xn(2))k2(p;q)k2(p;q)k1(p;q)k2tm.

This evidently completes the proof of Lemma 3.1. ∎

Lemma 3.2.

For each t[0,1] and nN,

Ln,p,qx(1),x(2)(f1;t)=txn(2)(f1(t)=t).

Proof.

Let t[0,1] be fixed. Then we easily find from (3.1) that

Ln,p,qx(1),x(2)(f1;t)=(j=12(p2-txn(j)q2)n)m=0[k1+k2=m(pα1;qα1)k1(pα2;qα2)k2f((p;q)k2n+(p;q)k2-1)(xn(1))k1(p;q)k1(xn(2))k2(p;q)k2(p;q)k1(p;q)k2]tm=(j=12(p2-txn(j)q2)n)m=1[k=1m(pα1;qα1)m-k(pα2;qα2)kf((p;q)kn+(p;q)k-1)(xn(1))m-k(p;q)m-k(xn(2))k(p;q)k(p;q)m-k(p;q)k]tm=txn(2)(j=12(p2-txn(j)q2)n)m=1[k=1m(pα1;qα1)m-k(pα2;qα2)k-1(xn(1))m-k(p;q)m-k(xn(2))k-1(p;q)k-1(p;q)m-k(p;q)k-1]tm-1=txn(2)(j=12(p2-txn(j)q2)n)m=1[k1+k2=m-1(pα1;qα1)k1(pα2;qα2)k2(xn(1))k1(p;q)k1(xn(2))k2(p;q)k2(p;q)k1(p;q)k2]tm-1.

Thus, clearly, we have

Ln,p,qx(1),x(2)(f1;t)=txn(2)j=12(p2-txn(j)q2)nm=1gm-1,p,q(α1,α2)(xn(1),xn(2))tm-1,

where

gm-1,p,q(α1,α2)(xn(1),xn(2))tm-1=k1+k2=m-1(pα1;qα1)k1(pα2;qα2)k2(xn(1))k1(p;q)k1(xn(2))k2(p;q)k2(p;q)k1(p;q)k2tm-1.

Now, by taking

r=2,α1=α2=n,x1=xn(1),x2=xn(2)

in (2.4), we have

Ln,p,qx(1),x(2)(f1;t)=txn(2),

which completes the proof of Lemma 3.2. ∎

Lemma 3.3.

For each t[0,1] and nN,

|Ln,p,qx(1),x(2)(f2;t)-f2(t)|2t2(1-xn(2))+txn(2)n(f2(t)=t2).

Proof.

Let t[0,1] be fixed. Then we easily find from (3.1) that

Ln,p,qx(1),x(2)(f2;t)=(j=12(p2-txn(j)q2)n)m=0[k1+k2=m(pα1;qα1)k1(pα2;qα2)k2f((p;q)k2n+(p;q)k2-1)2(xn(1))k1(p;q)k1(xn(2))k2(p;q)k2(p;q)k1(p;q)k2]tm=(j=12(p2-txn(j)q2)n)m=1[k=1m(pα1;qα1)m-k(pα2;qα2)kf((p;q)kn+(p;q)k-1)2(xn(1))m-k(p;q)m-k(xn(2))k(p;q)k(p;q)m-k(p;q)k]tm,

which yields

Ln,p,qx(1),x(2)(f2;t)=txn(2)(j=12(p2-txn(j)q2)n)m=1[k=1m(pα1;qα1)m-k(pα2;qα2)k-1f((p;q)kn+(p;q)k-1)(xn(1))m-k(p;q)m-k(xn(2))k-1(p;q)k-1(p;q)m-k(p;q)k-1]tm=t2(xn(2))2(j=12(p2-txn(j)q2)n)m=2[k=2m(pα1;qα1)m-k(pα2;qα2)k-2f(n+(p;q)k-2n+(p;q)k-1)(xn(1))m-k(p;q)m-k(xn(2))k-2(p;q)k-2(p;q)m-k(p;q)k-2]tm-2+txn(2)(j=12(p2-txn(j)q2)n)m=1[k=1m(pα1;qα1)m-k(pα2;qα2)k-1f(1n+(p;q)k-1)(xn(1))m-k(p;q)m-k(xn(2))k-1(p;q)k-1(p;q)m-k(p;q)k-1]tm-1.

Using this last representation, we get

Ln,p,qx(1),x(2)(f2;t)t2(xn(2))j=12(p2-txn(j)q2)nm=2gm-2,p,q(n,n)tm-2+txn(2)nj=12(p2-txn(j)q2)nm=2gm-1,p,q(n,n)tm-1=t2(xn(2))2+txn(2)n,

where

gm-2,p,q(n,n)(xn(1),xn(2))tm-2=k1+k2=m-2(pn;qn)k1(pn;qn)k2(xn(1))k1(p;q)k1(xn(2))k2(p;q)k2(p;q)k1(p;q)k2tm-2

and

gm-1,p,q(n,n)(xn(1),xn(2))tm-1=k1+k2=m-1(pn;qn)k1(pn;qn)k2(xn(1))k1(p;q)k1(xn(2))k2(p;q)k2(p;q)k1(p;q)k2tm-1,

which implies that

Ln,p,qx(1),x(2)(f2;t)-f2(t)t2((xn(2))2-1)+txn(2)n.(3.2)

On the other hand, since

0Ln,p,qx(1),x(2)((y-t)2;t)=Ln,p,qx(1),x(2)(f2;t)-2tLn,p,qx(1),x(2)(f1;t)+t2,

it follows from Lemmas 3.1 and 3.2 that

Ln,p,qx(1),x(2)(f2;t)-f2(t)2t2(xn(2)-1).(3.3)

Combining (3.2) with (3.3), we have

|Ln,p,qx(1),x(2)(f2;t)-f2(t)|2t2(1-xn(2))+txn(2)n,

which is precisely the inequality asserted by Lemma 3.3. ∎

We now present below a new Korovkin-type approximation theorem via the deferred weighted 𝒜-statistical convergence based upon our operators defined in (3.1).

Theorem 3.4.

Let A=(an,k) be a non-negative deferred weighted regular matrix. Also let (an) and (bn) be sequences of non-negative integers. Then

stat𝒜D(N¯)limnxn(2)=1(3.4)

if and only if, for all fC[0,1],

stat𝒜D(N¯)limnLn,p,qx(1),x(2)(f)-f=0.(3.5)

Proof.

First of all, we assume that assertion (3.5) holds true for all fC[0,1]. Then we have

stat𝒜D(N¯)limnLn,p,qx(1),x(2)(f1)-f1=0(f1C[0,1]).(3.6)

By applying Lemma 3.2, since

Ln,p,qx(1),x(2)(f1)-f1=1-xn(2),(3.7)

from (3.6) and (3.7) we immediately get

stat𝒜D(N¯)limnxn(2)=1.

Conversely, we assume that (3.4) holds true. Then it is easy to verify from Lemma 3.1 that

limnLn,p,qx(1),x(2)(f0)-f0=0(in the ordinary sense).

Moreover, since every convergent sequence is deferred weighted 𝒜-statistical convergent to the same value for any non-negative regular matrix 𝒜=(an,k), we get

stat𝒜D(N¯)limnLn,p,qx(1),x(2)(f0)-f0=0(f0(t)=1).(3.8)

It also follows from Lemma 3.2 that

Ln,p,qx(1),x(2)(f1)-f1=1-xn(2).

Since

stat𝒜D(N¯)limnxn(2)=1,

we may write

stat𝒜D(N¯)limnLn,p,qx(1),x(2)(f1)-f1=0(f1(t)=t).(3.9)

We now claim that

stat𝒜D(N¯)limnLn,p,qx(1),x(2)(f2)-f2=0(f2(t)=t2).(3.10)

Indeed, in view of Lemma 3.3, we have

limnLn,p,qx(1),x(2)(f2)-f22(1-xn(2))+xn(2)n.(3.11)

For a given ϵ>0, we now define the following sets:

B={n:Ln,p,qx(1),x(2)(f2)-f2ϵ},B1={n:1-xn(2)ϵ4},B2={n:xn(2)nϵ2}.

Thus, from (3.11) it is easy to see that BB1B2. Then, for each j, we get

nBaj,nnB1aj,n+nB2aj,n.(3.12)

Since

stat𝒜D(N¯)limnxn(2)=1,

we observe that

stat𝒜D(N¯)limn(xn(2)-1)=0

and

stat𝒜D(N¯)limnxn(2)n=0.

Using these facts and also by taking the limit when (n) in (3.12), we conclude that

limjnBaj,n=0(j=1,2),

which yields (3.10). Now, by combining (3.8), (3.9) and (3.10), and using a known result [14, Theorem 1] (see also [10] and [25]), we complete the proof of Theorem 3.4. ∎

Remark 3.5.

The above-cited known results [14, 10, 25] are given for statistical convergence and 𝒜-statistical convergence, but the present proof works also for deferred weighted 𝒜-statistical convergence.

In a similar manner, we can extend Theorem 3.6 to the r-dimensional case for the operators Ln,p,qx(1),,x(r)(f;t) given by (2.4) as follows.

Theorem 3.6.

Let A=(an,k) be a non-negative deferred weighted regular matrix. Also let (an) and (bn) be sequences of non-negative integers. Then

stat𝒜D(N¯)limnxn(r)=1

if and only if, for all fC[0,1],

stat𝒜D(N¯)limnLn,p,qx(1),,x(r)(f)-f=0.

Remark 3.7.

If, in Theorem 3.6, we put an=0, bn=n, sn=1 and p=1, then it simply reduces to the 𝒜-statistical convergence case (see [25, p. 6915, Theorem 2]). Furthermore, for an=0, bn=n and sn=1, and upon replacing 𝒜=(an,k) by the identity matrix I, we immediately get the following theorem, which is simply the ordinary (classical) convergence case of Theorem 3.6.

Theorem 3.8.

In the ordinary sense,

limnxn(r)=1

if and only if, for all fC[0,1], the sequence of positive linear operators

limnLn,p,qx(1),,x(r)(f)-f=0

is uniformly convergent to f on [0,1].

We now present below an illustrative example for the sequence of positive linear operators that neither satisfies the hypotheses of Theorem 3.8 nor the results of Erkuş et al. (see [10, p. 219, Theorem 2] and Mursaleen et al. [25, p. 6915, Theorem 2]), but it does satisfy the hypotheses of our Theorem 3.6.

Example 3.9.

Let us consider 𝒜 as a Cesàro matrix defined as follows:

an,k={1n,1kn,0,k>n.

For an=2n, bn=4n and sn=1, it can easily be seen that 𝒜w+. Furthermore, we consider the sequence x(j)=(xn(j))n (j=1,2,,r-1) such that 0<xn(j)<1 is given by

xn(r)={12,n=m2,m,nn+1otherwise.(3.13)

Here, in this example, we observe that

0<xn(r)<1(n)

and also that

stat𝒜D(N¯)limnxn(r)=1.

Therefore, by Theorem 3.6, for all fC[0,1] we have

stat𝒜D(N¯)limnLn,p,qx(1),,x(r)(f)-f=0.

However, since the sequence (xn(r)) defined by (3.13) is non-convergent (in the ordinary sense), Theorem 3.8 does not hold true in this case, while our Theorem 3.6 still works. Furthermore, (xn(r)) defined by (3.13) is not 𝒜-statistical convergent, that is, the results of Erkuş et al. [10, p. 219, Theorem 2] and Mursaleen et al. [25, p. 6915, Theorem 2] do not hold true in this case, while our Theorem 3.6 still works. Thus, clearly, our Theorem 3.6 is stronger than each of these earlier results.

4 Rates of the deferred weighted 𝒜-statistical convergence

In this section, we study the rates of the deferred weighted 𝒜-statistical convergence for our sequence of positive linear operators in Theorem 3.6 with the help of the modulus of continuity and the elements of the Lipschitz class.

Let fC[0,1]. The modulus of continuity of f denoted by ω(f,δ) is defined by

ω(f,δ)=sup|y-t|<δt,y[0,1]|f(y)-f(t)|.

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. A necessary and sufficient condition for a function fC[0,1] is that

limδ0ω(f,δ)=0.

It is also well known that, for any δ>0 and for each t,y[0,1],

|f(y)-f(t)|(|y-t|δ+1)ω(f,δ).(4.1)

Now we state and prove a new result in the form of the following theorem.

Theorem 4.1.

For each nN and for all fC[0,1],

Ln,p,qx(1),,x(r)(f)-f2ω(f,δn)(4.2)

holds, where

δn=(xn(r)n+4(1-xn(r)))12.(4.3)

Proof.

Let fC[0,1] and let t[0,1] be fixed. By linearity and monotonicity of Ln,p,qx(1),,x(r)(f;t), and by using (4.1), we find for any δ>0 that

|Ln,p,qx(1),,x(r)(f;t)-f(t)|Ln,p,qx(1),,x(r)(|f(y)-f(t)|;t)ω(f,δ)Ln,p,qx(1),,x(r)(1+|y-t|δ;t)ω(f,δ)(1+1δLn,p,qx(1),,x(r)(|y-t|;t)).

Furthermore, by the Cauchy–Schwarz inequality for positive linear operators, we have

|Ln,p,qx(1),,x(r)(f;t)-f(t)|ω(f,δ){1+1δ(Ln,p,qx(1),,x(r)(ψ;t))12},(4.4)

where

ψ=ψ(y)=(y-t)2.

We also observe that

Ln,p,qx(1),,x(r)(ψ;t)=Ln,p,qx(1),,x(r)(f2;t)-2xLn,p,qx(1),,x(r)(f1;t)+t2|Ln,p,qx(1),,x(r)(f2;t)-f2(t)|+2x|Ln,p,qx(1),,x(r)(f1;t)-f1(t)|,

which, by virtue of Lemmas 3.2 and 3.3, yields

Ln,p,qx(1),,x(r)(ψ;t)2t2(1-xn(r))+txn(r)n+2t2(1-xn(r))txn(r)n+4t2(1-xn(r)).

Now, combining this last inequality with (4.4), we get

|Ln,p,qx(1),,x(r)(f;t)-f(t)|ω(f,δ){1+1δ(txn(r)n+4t2(1-xn(r)))12},

which implies that

Ln,p,qx(1),,x(r)(f;t)-f(t)ω(f,δ){1+1δ(xn(r)n+4(1-xn(r)))12}.(4.5)

If we choose δ=δn, where δn is given as in (4.3), then assertion (4.2) of Theorem 3.8 follows immediately from (4.5). ∎

We now turn to our investigation of the rate of convergence of the positive linear operators Ln,p,qx(1),,x(r)(f;t) given by (2.4) by means of the elements of the Lipschitz class LipM(α) (0<α1). We first recall that a function fC[0,1] belongs to the Lipschitz class LipM(α) (0<α1) if the following inequality holds true:

|f(y)-f(t)|M|y-t|α(t,y[0,1], 0<α1,M>0).(4.6)

We now state and prove a new result in the form of the following theorem.

Theorem 4.2.

For each nN and for all fLipM(α),

Ln,p,qx(1),,x(r)(f;t)-f(t)2δnα,(4.7)

where δn is the same as in Theorem 3.8.

Proof.

Let fLipM(α) and suppose that 0t1. We then find from (4.6) that

Ln,p,qx(1),,x(r)(f;t)-f(t)Ln,p,qx(1),,x(r)(|f(y)-f(t)|;t)MLn,p,qx(1),,x(r)(|y-t|α;t).

Now, by applying Hölder’s inequality with

p=2αandq=22-α,

we get

Ln,p,qx(1),,x(r)(f;t)-f(t)M(Ln,p,qx(1),,x(r)(ψ;t))α2,(4.8)

where

ψ=ψ(y)=(y-t)2.

Just as in the proof of Theorem 4.1, since

Ln,p,qx(1),,x(r)(ψ;t)txn(r)n+4t2(1-xn(r)),

we can deduce from (4.8) that

|Ln,p,qx(1),,x(r)(f;t)-f(t)|M(txn(r)n+4t2(1-xn(r)))α2,

which readily yields

|Ln,p,qx(1),,x(r)(f;t)-f(t)|M(xn(r)n+4(1-xn(r)))α2.(4.9)

If we take (δn) as in (4.3), then assertion (4.7) of Theorem 4.2 would follow from this last inequality (4.9). ∎

5 Concluding remarks and observations

In this concluding section of our investigation, we present several further remarks and observations concerning the various results which we have proved here.

Remark 5.1.

Let (xn(r))n be the sequence given in Example 3.9. Then, since

stat𝒜D(N¯)limnxn(r)=1on C[0,1],

by applying Theorem 3.6, we can write

stat𝒜D(N¯)limnLn,p,qx(1),,x(r)(f)-fC[0,1]=0,(fC[0,1]).(5.1)

However, since (xn(r)) is not ordinarily convergent and so it does not also converge uniformly in the ordinary sense, the classical Korovkin Theorem (Theorem 3.8) does not work here for the operators defined by (2.4). Hence, this application clearly indicates that our Theorem 3.6 is a non-trivial generalization of the classical Korovkin-type theorem (see Theorem 3.8).

Remark 5.2.

Let (xn(r))n be the sequence as given in Example 3.9. Then, since

stat𝒜D(N¯)limnxn(r)=1on C[0,1],

by applying our Theorem 3.6, condition (5.1) holds true. However, since (xn(r)) is not 𝒜-statistically convergent, we can say that the results of Erkuş et al. [10, p. 219, Theorem 2] and Mursaleen et al. [25, p. 6915, Theorem 2] do not hold true for our operators defined in (2.4). Thus, our Theorem 3.6 is also a non-trivial extension of the results of Erkuş et al. [10, p. 219, Theorem 2] and Mursaleen et al. [25, p. 6915, Theorem 2]. Based upon the above results, it is concluded here that our proposed method has successfully worked for the operators defined in (2.4), and therefore that our results are stronger than the classical and 𝒜-statistical versions of the Korovkin-type approximation theorem established earlier (see [10, 25]).

Remark 5.3.

For a non-negative deferred weighted regular matrix A=(an,k), let (an) and (bn) be sequences of non-negative integers. Then, if we first set

stat𝒜D(N¯)limnxn(r)=1,

we find that

stat𝒜D(N¯)limnδn=0,

where δn is given by (4.3). It also implies that

stat𝒜D(N¯)limnω(f,δn)=0(fC[0,1]).

Thus, clearly, Theorems 4.1 and 4.2 give us the rates of deferred weighted 𝒜-statistical convergence in Theorem 3.6. Of course, if we replace the matrix 𝒜=(an,k) by the identity matrix I and let (an)=0 and (bn)=n, then we naturally get the corresponding rates of classical (ordinary) convergence in Theorem 3.8.

Remark 5.4.

In our present investigation, we have considered a number of interesting special cases and illustrative examples in support of our definitions and also of the results which we have established here.

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About the article

Received: 2018-02-24

Accepted: 2018-04-12

Published Online: 2018-05-03

Published in Print: 2018-06-01


Citation Information: Journal of Applied Analysis, Volume 24, Issue 1, Pages 1–16, ISSN (Online) 1869-6082, ISSN (Print) 1425-6908, DOI: https://doi.org/10.1515/jaa-2018-0001.

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