# Some Approximation Properties ofq-Baskakov-Beta-Stancu Type Operators

Authors: Vishnu Narayan Mishra; Kejal Khatri; Lakshmi Narayan Mishra
Journal: Journal of Calculus of Variations
 (2013)

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## Abstract

This paper deals with new typeq-Baskakov-Beta-Stancu operators defined in the paper. First, we have used the properties of q-integral to establish the moments of these operators. We also obtain some approximation properties and asymptotic formulae for these operators. In the end we have also presented better error estimations for the q-operators.


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## Body

## 1. Introduction

In the recent years, the quantum calculus (q-calculus) has attracted a great deal of interest because of its potential applications in mathematics, mechanics, and physics. Due to the applications of q-calculus in the area of approximation theory, q-generalization of some positive operators has attracted much interest, and a great number of interesting results related to these operators have been obtained (see, for instance, [1–3]). In this direction, several authors have proposed the q-analogues of different linear positive operators and studied their approximation behaviors. Also, Aral and Gupta [4] defined q-generalization of the Baskakov operators and investigated some approximation properties of these operators. Subsequently, Finta and Gupta [5] obtained global direct error estimates for these operators using the second-order Ditzian Totik modulus of smoothness. To approximate Lebesgue integrable functions on the interval [0,∞), modified Beta operators [6] are defined as
(1)Bn(f,x)=n-1n∑k=0∞‍bn,k(x)∫0∞‍pn,k(t)f(t)dt,x∈[0,∞),
where bn,k(x)=(1/B(k+1,n))(xk/(1+x)n+k+1) and pn,k(t)=(n+k+1k)(tk/(1+t)n+k).The discreteq-Beta operators are defined as
(2)Vnq(f,x)=1[n]q∑k=0∞‍bn,kq(x)f([k]q[n+1]qqk-1).Recently, Maheshwari and Sharma [7] introduced the q-analogue of the Baskakov-Beta-Stancu operators and studied the rate of approximation and weighted approximation of these operators. Motivated by the Stancu type generalization of q-Baskakov operators, we propose the q-analogue of the operators Bn(α,β), recently introduced and studied for special values α=β=0 by Gupta and Kim [8] as
(3)Bn,α,βq(f,x)∶=[n-1]q[n]q∑k=0∞‍bn,kq(x)×∫0∞/A‍qkpn,kq(t)f([n]qt+α[n]q+β)dqt,
where bn,kq(x)=(qk(k-1)/2/Bq(k+1,n))(xk/(1+x)qn+k+1) and pn,kq(t)=(n+k-1k)q(qk(k-1)/2/Bq(k+1,n))(tk/(1+t)qn+k+1).We know that∑k=0∞‍bn,kq(x)=[n]q and ∫0∞/A‍qkpn,kq(t)=1/[n-1]q. We mention that Bn,0,0q≡Bnq (see [8]).Very recently, Gupta et al. [9] introduced some direct results in simultaneous approximation for Baskakov-Durrmeyer-Stancu operators. The aim of this paper is to study the approximation properties of a new generalization of the Baskakov type Beta Stancu operators based on q-integers. We estimate moments for these operators. Also, we study asymptotic formula for these operators. Finally, we give better error estimations for the operator (3). First, we recall some definitions and notations of q-calculus. Such notations can be found in [10, 11]. We consider q as a real number satisfying q>0. For n∈ℕ,
(4)[n]q∶={1-qn1-q,q≠1,n,q=1,[n]q!∶={[n]q[n-1]q[n-2]q,…,[1]q,n=1,2,…,1,n=0,(1+x)qn∶=∏j=0n-1‍(1+qjx).
The q-binomial coefficients are given by
(5)(nk)q=[n]q![k]q![n-k]q!,0≤k≤n.
The q-derivative 𝒟qf of a function f is given by
(6)(𝒟qf)(x)=f(x)-f(qx)(1-q)x,ifx≠0.
The q-analogues of product and quotient rules are defined as
(7)𝒟q(f(x)g(x))=g(x)𝒟qf(x)+f(qx)𝒟qg(x),𝒟q(f(x)g(x))=g(x)𝒟qf(x)-f(x)𝒟qg(x)g(x)g(qx).
The q-Jackson integrals and the q-improper integrals are defined as [12, 13]
(8)∫0a‍f(x)dqx=(1-q)a∑n=0∞‍f(aqn)qn,a>0,(9)∫0∞/A‍f(x)dqx=(1-q)∑n=-∞∞‍f(qnA)qnA,A>0,
provided that the sums converge absolutely. Using (9), De Sole and Kac [14] defined the q-analogue of Beta functions of second kind B(t,s)=∫0∞‍(xt-1/(1+x)t+s)dx as follows:
(10)Bq(t,s)=K(A,t)∫0∞/A‍xt-1(1+x)qt+sdqx,
where K(x,t)=(1/(1+x))xt(1+1/x)qt(1+x)q1-t. This function is q-constant in x; that is, K(qx,t)=K(x,t). It was observed in [14] that Bq(t,s) is independent of A; this is because from the integral and K(A,t) the term A cancels out. In particular for any positive integer n, we have
(11)K(x,n)=qn(n-1)/2,K(x,0)=1.
Also, we have
(12)Bq(t,s)=[t-1]q![s-1]q![t+s-1]q!.
In [8], Gupta and Kim obtained recurrence formula for the moments of the operators as follows.Theorem 1 (see [8]).
If one defined the central moments as(13)Tn,m(x)≔Bnq(tm;x)=[n-1]q[n]q∑k=0∞‍bn,kq(x)×∫0∞/A‍qkpn,kq(t)tmdqt,
then, for n>m+2, one has the following recurrence relation:
(14)([n]q-[m+2]q)Tn,m+1(qx)=qx(1+x)Dq[Tn,m(x)]+q([m+1]q+[n+1]qx)Tn,m(qx).


## 2. Moment Estimates

Lemma 2 (see [8]).
The following equalities hold.(i)
Bnq(1,x)=1.(ii)
Bnq(t,x)=([n+1]q/q2[n-2]q)x+(1/q[n-2]q).(iii)
Bnq(t2,x)=([n+1]q[n+2]q/q6[n-2]q[n-3]q)x2+([n+1]q[2]q2/q5[n-2]q[n-3]q)x+([2]q/q3[n-2]q[n-3]q), for n>3.Lemma 3.
The following equalities hold.(i)
Bn,α,βq(1,x)=1.(ii)
Bn,α,βq(t,x)=([n+1]q[n]q/q2[n-2]q([n]q+β))x+([n]q/q[n-2]q+α)(1/([n]q+β)), for n>2.(iii)
Bn,α,βq(t2,x)=([n+1]q[n+2]q[n]q2/q6[n-2]q[n-3]q([n]q+β)2)x2+([n]q[2]q2/q3[n-3]q+2α)([n+1]q[n]q/q2[n-2]([n]q+β)2)x+[2]q[n]q2/q3[n-2]q[n-3]q([n]q+β)2+2α[n]q/q[n-2]q([n]q+β)2+α2/([n]q+β)2, for n>3.Proof.
The operatorsBn,α,βq are well defined on function 1, t, t2. By Lemma 2, for every n>0 and x∈[0,∞), we have
(15)Bn,α,βq(1,x)=Bnq(1,x)=1,Bn,α,βq(t,x)=[n-1]q[n]q∑k=0∞‍bn,kq(x)×∫0∞/A‍qkpn,kq(t)([n]qt+α[n]q+β)dqt=[n]q([n]q+β)Bnq(t,x)+α([n]q+β)Bnq(t,x)=[n+1]q[n]qq2[n-2]q([n]q+β)x+([n]qq[n-2]q+α)1([n]q+β).
Similarly,
(16)Bn,α,βq(t2,x)=[n-1]q[n]q∑k=0∞‍bn,kq(x)×∫0∞/A‍qkpn,kq(t)([n]qt+α[n]q+β)2dqt=([n]q[n]q+β)2Bnq(t2,x)+2α[n]q([n]q+β)2Bnq(t,x)+(α[n]q+β)2Bnq(1,x)=[n+1]q[n+2]q[n]q2q6[n-2]q[n-3]q([n]q+β)2x2+([n]q[2]q2q3[n-3]q+2α)×[n+1]q[n]qq2[n-2]([n]q+β)2x+[2]q[n]q2q3[n-2]q[n-3]q([n]q+β)2+2α[n]qq[n-2]q([n]q+β)2+α2([n]q+β)2.Remark 4.
For allm∈{0,1,2,…}, 0≤α≤β, we have the following recursive relation for the images of the monomials tm under Bn,α,βq(tm,x) in terms of Bnq(tr,x), r=0,1,2,…,m, as
(17)Bn,α,βq(tm,x)=∑r=0m‍(mr)nrαm-r(n+β)mBnq(tr,x).Remark 5.
If we putq=1 and α=β=0, we get the moments of the modified Beta operators [6] as
(18)Bn1(t,x)=(n+1)x+1n-2,n>2,Bn1(t2,x)=(n+1)(n+2)x2+4(n+1)x+2(n-2)(n-3),n>3.Remark 6.
From Lemma3, we have
(19)Dn,α,β(x)=Bn,α,βq((t-x),x)=([n+1]q[n]qq2[n-2]q([n]q+β)-1)x+([n]qq[n-2]q+α)1([n]q+β),forn>2,En,α,β(x)=Bn,α,βq((t-x)2,x)=Bn,α,βq(t2,x)-2xBn,α,βq(t,x)+x2=([n+1]q[n+2]q[n]q2q6[n-2]q[n-3]q([n]q+β)2-2[n+1]q[n]qq2[n-2]q([n]q+β)+1[n+1]q[n+2]q[n]q2q6[n-2]q[n-3]q([n]q+β)2)x2+(∑ttt[n+1]q[n]q2[2]q2q5[n-2]q[n-3]q([n]q+β)2+2[n+1]q[n]qαq2[n-2]q([n]q+β)2-2([n]qq[n-2]q+α)1([n]q+β)∑ttt)x+[2]q[n]q2q3[n-2]q[n-3]q([n]q+β)2+2α[n]qq[n-2]q([n]q+β)2+α2([n]q+β)2,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiforn>3.


## 3. Direct Result and Asymptotic Formula

Let the spaceCB[0,∞) of all real-valued continuous bounded functions be endowed with the norm ∥f∥=sup{|f(t)|:x∈[0,∞)}. Further, let us consider the following K-functional:
(20)K2(f;δ)=infg∈C2[0,∞){∥f-g∥+δ∥g′′∥},
where δ>0 and CB2={g∈CB[0,∞):g′,g′′∈CB[0,∞)}. By [15, page 177, Theorem  2.4], there exists an absolute constant M>0 such that
(21)K2(f;δ)≤Mω2(f;δ),
where
(22)ω2(f;δ)=sup0<h≤δ(supx∈[0,∞)|f(x+2h)-2f(x+h)+f(x)|)
is the second-order modulus of smoothness of f∈CB[0,∞). Also we set
(23)ω(f;δ)=sup0<h≤δ(supx∈[0,∞)|f(x+h)-f(x)|).Theorem 7.
Letf∈CB[0,∞) and q=qn∈(0,1) such that qn→1 as n→∞. Then for all x∈[0,∞) and n>3, there exists an absolute constant C>0 such that
(24)|Bn,α,βq(f;x)-f(x)|≤Cω2(f;En,α,β(x)+Dn,α,β2(x))+ω(f;|Dn,α,β(x)|).Proof.
We are introducing the auxiliary operators as follows:(25)B^n,α,βq(f;x)=Bn,α,βq(f;x)+f(x)-f(∑ttt[n+1]q[n]qq2[n-2]q([n]q+β)∑ttt×(x+q2α[n-2]q[n]q[n+1]q+q[n+1]q)).
From (25) and Lemma 3, we have
(26)B^n,α,βq(t-x;x)=0.
Let x∈[0,∞) and g∈CB2[0,∞). Using Taylor’s formula
(27)g(t)-g(x)=(t-x)g′(x)+∫xt‍(t-u)g′′(u)du,
applying B^n,α,βq, and by (26), we get
(28)B^n,α,βq(g;x)-g(x)=B^n,α,βq((t-x)g′(x);x)+B^n,α,βq(∫xt‍(t-u)g′′(u)du;x)=g′(x)B^n,α,βq((t-x);x)+Bn,α,βq(∫xt‍(t-u)g′′(u)du;x)-∫x([n+1]q[n]q/q2[n-2]q([n]q+β))x+([n]q/q[n-2]q+α)(1/([n]q+β))‍(∑ttt[n+1]q[n]qq2[n-2]q([n]q+β)x+([n]qq[n-2]q+α)1([n]q+β)-u∑ttt)g′′(u)du=Bn,α,βq(∫xt‍(t-u)g′′(u)du;x)-∫x[n+1]q[n]qx/q2[n-2]q([n]q+β)+([n]q/q[n-2]q+α)(1/([n]q+β))‍(∑ttt[n+1]q[n]qxq2[n-2]q([n]q+β)+([n]qq[n-2]q+α)1([n]q+β)-u∑ttt)g′′(u)du.
On the other hand, since
(29)|∫xt‍|t-u||g′′(u)|du|≤∥g′′∥(t-x)2,|∑ttt∫x[n+1]q[n]qx/q2[n-2]q([n]q+β)+([n]q/q[n-2]q+α)(1/([n]q+β))‍(∑ttt[n+1]q[n]qxq2[n-2]q([n]q+β)+([n]qq[n-2]q+α)×1([n]q+β)-u∑ttt)g′′(u)du|≤∥g′′∥(∑ttt[n+1]q[n]qxq2[n-2]q([n]q+β)+([n]qq[n-2]q+α)1([n]q+β)-x∑ttt)2=Dn,α,β2(x)∥g′′∥.
We conclude by Remark 6 that
(30)|B^n,α,βq(g;x)-g(x)|=|Bn,α,βq(∫xt‍(t-u)g′′(u)du;x)|+|∑ttt∫x[n+1]q[n]qx/q2[n-2]q([n]q+β)+([n]q/q[n-2]q+α)(1/([n]q+β))‍(∑ttt[n+1]q[n]qxq2[n-2]q([n]q+β)+([n]qq[n-2]q+α)1([n]q+β)-u∑ttt)g′′(u)du|≤Bn,α,βq((t-x)2∥g′′∥;x)+(∑ttt[n+1]q[n]qxq2[n-2]q([n]q+β)+([n]qq[n-2]q+α)1([n]q+β)-x∑ttt)2∥g′′∥(31)|B^n,α,βq(g;x)-g(x)|=En,α,β(x)∥g′′∥+Dn,α,β2(x)∥g′′∥.
From (25), we can write that
(32)|Bn,α,βq(f;x)-f(x)|≤|B^n,α,βq(f;x)-f(x)|+|f(x)-f(∑ttt[n+1]q[n]qq2[n-2]q([n]q+β)×(x+q2α[n-2]q[n]q[n+1]q+q[n+1]q)∑ttt)|≤|B^n,α,βq(f-g;x)-(f-g)(x)|+|B^n,α,βq(g;x)-g(x)|+|∑tttf(x)-f([n+1]q[n]qq2[n-2]q([n]q+β)×(x+q2α[n-2]q[n]q[n+1]q+q[n+1]q)∑ttt)|.
Now, taking into account the boundedness of Bn,α,βq and from (31), we get
(33)|Bn,α,βq(f;x)-f(x)|≤4∥f-g∥+(En,α,β(x)+Dn,α,β2(x))∥g′′∥+|f(x)-f(∑ttt[n+1]q[n]qq2[n-2]q([n]q+β)×(x+q2α[n-2]q[n]q[n+1]q+q[n+1]q)∑ttt)|≤4∥f-g∥+ω(f;|Dn,α,β(x)|)+(En,α,β(x)+Dn,α,β2(x))∥g′′∥.
Now, taking infimum on the right-hand side over all g∈CB2[0,∞) and from (21), we get
(34)|Bn,α,βq(f;x)-f(x)|≤4K2(f;En,α,β(x)+Dn,α,β2(x))+ω(f;|Dn,α,β(x)|)=Cω2(f;En,α,β(x)+Dn,α,β2(x))+ω(f;|Dn,α,β(x)|),
where 4C1=C>0. This proves the theorem.Our next result in this section is an asymptotic formula.Theorem 8.
Letf be bounded and integrable function on the interval [0,∞); the second derivative of f exists at a fixed point x∈[0,∞) and q=qn∈(0,1) such that qn→1 as n→∞. Consider
(35)limn→∞[n]qn[Bn,α,βqn(f;x)-f(x)]=((3-β)x+(1+α))f′(x)+x(1+x)f′′(x).Proof.
Using Taylor’s expansion off, we can write
(36)f(t)-f(x)=(t-x)f′(x)+(t-x)22!f′′(x)+ε(t-x)(t-x)2,
whereε is bounded and limt→0ε(t)=0. Applying the operator Bn,α,βqn to the above relation, we get
(37)Bn,α,βqn(f;x)-f(x)=f′(x)Bn,α,βqn((t-x);x)+f′′(x)2!Bn,α,βqn((t-x)2;x)+Bn,α,βqn(ε(t-x)(t-x)2;x)=f′(x)Dn,α,β(x)+f′′(x)2!En,α,β(x)+Bn,α,βqn(ε(t-x)(t-x)2;x),
where Dn,α,β(x) and En,α,β(x) are defined in Remark 6.
Using Cauchy-Schwarz inequality, we have(38)[n]qnBn,α,βqn(ε(t-x)(t-x)2;x)≤Bn,α,βqn(ε2(t-x);x)1/2([n]qn2Bn,α,βqn((t-x)4;x))1/2.
Using Theorem 1 with the help of Remark 4, we can easily find that
(39)limn→∞[n]qn2Bn,α,βqn((t-x)4;x)=0.
Also, since
(40)limn→∞[n]qnDn,α,β(x)=(3-β)x+(1+α),limn→∞[n]qnEn,α,β=2x(1+x).
Thus,
(41)limn→∞[n]qn[Bn,α,βqn(f;x)-f(x)]=((3-β)x+(1+α))f′(x)+x(1+x)f′′(x),
which completes the proof.


## 4. Better Estimation

It is well known that the operators preserve constant as well as linear functions. To make the convergence faster, King [16] proposed an approach to modify the classical Bernstein polynomials, so that this sequence preserves two test functions: e0 and e2. After this, several researchers have studied that many approximating operators, L, possess these properties; that is, L(ei;x)=ei(x), where ei(x)=xi  (i=0,1) or xi  (i=0,2), for example, Bernstein, Baskakov, and Baskakov-Durrmeyer-Stancu operators (see [4, 5, 17–19]).As the operatorsBn,α,βq introduced in (3) preserve only the constant functions, further modification of these operators is proposed to be made so that the modified operators preserve the constant as well as linear functions. For this purpose, the modification of Bn,α,βq is as follows:
(42)B~n,α,βq(f;x)=Bn,α,βq(f;rn,q(x))=[n-1]q[n]q∑k=0∞‍bn,kq(rn,q(x))×∫0∞/A‍qkpn,kq(t)f([n]qt+α[n]q+β)dqt,
where
(43)rn,q(x)=(-q[n]q-αq2[n-2]q+q2x[n-2]q([n]q+β)[n]q[n+1]q),x∈In,q=[[n]qq[n-2]q([n]q+β)+α([n]q+β),∞).Lemma 9.
For eachx∈In,q, one has
(44)B~n,α,βq(1,x)=1,B~n,α,βq(t,x)=x,B~n,α,βq(t2,x)=[n-2]q[n+2]qq2[n+1]q[n-3]qx2+1q3[n-3]([n]q+β)×(∑ttt[n]q[2]q2+2q3α[n-3]q-2[n+2]q([n]q+qα[n-2]q)[n+1]q∑ttt)x+1q4[n-2]q[n-3]q([n]q+β)2×(∑ttt[n+2]q([n]q+qα[n-2]q)2[n+1]q-[n]q2[2]q-qα[2]q2[n]q[n-2]q-q4α2[n-3]q[n-2]q∑ttt),forn>3.Lemma 10.
For eachx∈In,q, the following equalities hold:
(45)B~n,α,βq((t-x),x)=0,Fn,α,β=B~n,α,βq((t-x)2,x)=([n-2]q[n+2]qq2[n+1]q[n-3]q-1)x2+1q3[n-3]([n]q+β)×(∑ttt[n]q[2]q2-2[n+2]q([n]q+qα[n-2]q)[n+1]qkkkk+2q3α[n-3]q∑ttt)x+1q4[n-2]q[n-3]q([n]q+β)2×(∑ttt[n+2]q([n]q+qα[n-2]q)2[n+1]qkkkkk-[n]q2[2]q-qα[2]q2[n]q[n-2]qkkkkk-q4α2[n-3]q[n-2]q∑ttt),forn>3.Theorem 11.
Letf∈CB(In,q) and q=qn∈(0,1) such that qn→1 as n→∞. Then for all x∈In,q and n>3, there exists an absolute constant C>0 such that
(46)|B~n,α,βq(f;x)-f(x)|≤Cω2(f;Fn,α,β(x)).Proof.
Letx,t∈In,q and g∈CB2(In,q). Using Taylor’s formula, we get
(47)g(t)-g(x)=(t-x)g′(x)+∫xt‍(t-u)g′′(u)du.
Applying B~n,α,βq, we get
(48)B~n,α,βq(g;x)-g(x)=g′(x)B~n,α,βq((t-x);x)+B~n,α,βq(∫xt‍(t-u)g′′(u)du;x).
Obviously, we have
(49)|∫xt‍|t-u||g′′(u)|du|≤∥g′′∥(t-x)2.
Therefore,
(50)|B~n,α,βq(g;x)-g(x)|≤∥g′′∥B~n,α,βq((t-x)2;x)=Fn,α,β∥g′′∥.
Since |B~n,α,βq(f;x)-f(x)|≤2∥f∥, thus
(51)|B~n,α,βq(f;x)-f(x)|≤|B~n,α,βq(f-g;x)-(f-g)(x)|+|B~n,α,βq(g;x)-g(x)|≤2∥f-g∥+Fn,α,β(x)∥g′′∥.
Now, taking infimum on the right-hand side over allg∈CB2(In,q) and from (21), we get
(52)|B~n,α,βq(f;x)-f(x)|≤Cω2(f;Fn,α,β(x)).
which proves the theorem.Theorem 12.
Letf be bounded and integrable function on the interval In,q; the second derivative of f exists at a fixed point x∈In,q and q=qn∈(0,1) such that qn→1 as n→∞; then
(53)limn→∞[n]qn[B~n,α,βqn(f;x)-f(x)]=x(1+x)f′′(x).The proof follows along the same lines of Theorem8.



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*Source: 814824-2013-12-16.xml*