This paper deals with new type q-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.
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 discrete q-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 EstimatesLemma 2 (see [8]).
The following equalities hold.
Bnq(1,x)=1.
Bnq(t,x)=([n+1]q/q2[n-2]q)x+(1/q[n-2]q).
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.
Bn,α,βq(1,x)=1.
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.
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 operators Bn,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 all m∈{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 put q=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 Lemma 3, 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 space CB[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.
Let f∈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.
Let f 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 of f, we can write
(36)f(t)-f(x)=(t-x)f′(x)+(t-x)22!f′′(x)+ε(t-x)(t-x)2,
where ε 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 operators Bn,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 each x∈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 each x∈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.
Let f∈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.
Let x,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 all g∈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.
Let f 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 Theorem 8.
Conflict of Interests
The authors declare that there is no conflict of interests.
Acknowledgments
The authors thank the anonymous learned reviewers for their valuable suggestions, which substantially improved the standard of the paper. Special thanks are due to Professor Adam Kowalewski, for kind cooperation and smooth behavior during communication and for his efforts to send the reports of the paper timely.
GovilN. K.GuptaV.Convergence of q-MeyerKonigZellerDurrmeyer operators20091997108GuptaV.A note on modified Baskakov type operators199410374782-s2.0-5124916871810.1007/BF02836820GuptaV.Some approximation properties of q-Durrmeyer operators200819711721782-s2.0-3894920061410.1016/j.amc.2007.07.056AralA.GuptaV.Generalized q-Baskakov operators20116146196342-s2.0-7995989229610.2478/s12175-011-0032-3FintaZ.GuptaV.Approximation properties of q-Baskakov operators2010811992112-s2.0-7604908557010.2478/s11533-009-0061-0GuptaV.AhmadA.Simultaneous approximation by modified Beta operators1996541122MaheshwariP.SharmaD.Approximation by q Baskakov-Beta-Stancu operators201261229730510.1007/s12215-012-0090-6GuptaV.KimT.On the rate of approximation by q modified Beta operators201137724714802-s2.0-7925162970210.1016/j.jmaa.2010.11.021GuptaV.VermaD. K.AgrawalP. N.Simultaneous approximation by certain Baskakov-Durrmeyer-Stancu operators201220318318710.1016/j.joems.2012.07.001GasperG.RahmanM.199035Cambridge, UKCambridge University PressEncyclopedia of Applied and Computational MathematicsKacV. G.CheungP.2002New York, NY, USAUniversitext, SpringerKimT.q-generalized Euler numbers and polynomials20061332932982-s2.0-3384528454610.1134/S1061920806030058KimT.Note on the Euler q-zeta functions20091297179818042-s2.0-6734924294610.1016/j.jnt.2008.10.007De SoleA.KacV. G.On integral representation of q-gamma and q-beta functions20051161129DeVoreR. A.LorentzG. G.1993Berlin, GermanySpringerKingJ. P.Positive linear operators which preserve x220039932032082-s2.0-003740704310.1023/A:1024571126455AgratiniO.Linear operators that preserve some test functions20061194136AgratiniO.On the iterates of a class of summation-type linear positive operators2008556117811802-s2.0-3884920103910.1016/j.camwa.2007.04.044VermaD. K.GuptaV.AgrawalP. N.Some approximation properties of Baskakov-Durrmeyer-Stancu operators201221811654965562-s2.0-8485590677810.1016/j.amc.2011.12.031