# Approximation by theq-Szász-Mirakjan Operators

Authors: N. I. Mahmudov
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper deals with approximating properties of theq-generalization of the Szász-Mirakjan operators in the case q>1. Quantitative estimates of the convergence in the polynomial-weighted spaces and the Voronovskaja's theorem are given. In particular, it is proved that the rate of approximation by the q-Szász-Mirakjan operators (q>1
) is of order q−n versus 1/n for the classical Szász-Mirakjan operators.


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## Body

## 1. Introduction

The approximation of functions by using linear positive operators introduced viaq-Calculus is currently under intensive research. The pioneer work has been made by Lupaş [1] and Phillips [2] who proposed generalizations of Bernstein polynomials based on the q-integers. The q-Bernstein polynomials quickly gained the popularity, see [3–11]. Other important classes of discrete operators have been investigated by using q-Calculus in the case 0<q<1, for example, q-Meyer-König operators [12–14], q-Bleimann, Butzer and Hahn operators [15–17], q-Szász-Mirakjan operators [18–21], and q-Baskakov operators [22, 23].In the present paper, we introduce aq-generalization of the Szász operators in the case q>1. Notice that different q-generalizations of Szász-Mirakjan operators were introduced and studied by Aral and Gupta [18, 19], by Radu [20], and by Mahmudov [21] in the case 0<q<1. Since we define q-Szász-Mirakjan operators for q>1, the rate of approximation by the q-Szász-Mirakjan operators (q>1) is of order q-n, which is essentially better than 1/n (rate of approximation for the classical Szász-Mirakjan operators). Thus our q-Szász-Mirakjan operators have better approximation properties than the classical Szász-Mirakjan operators and the other q-Szász-Mirakjan operators.The paper is organized as follows. In Section2, we give standard notations that will be used throughout the paper, introduce q-Szász-Mirakjan operators, and evaluate the moments of Mn,q. In Section 3 we study convergence properties of the q-Szász-Mirakjan operators in the polynomial-weighted spaces. In Section 4, we give the quantitative Voronovskaja-type asymptotic formula.


## 2. Construction ofMn,q and Estimation of Moments

Throughout the paper we employ the standard notations ofq-calculus, see [24, 25].q-integer and q-factorial are defined by
(2.1)[n]q∶={1-qn1-q,ifq∈R+∖{1},n,ifq=1,forn∈ℕ,[0]=0,[n]q!∶=[1]q[2]q…[n]q,forn∈ℕ,[0]!=1.
For integers 0≤k≤n  q-binomial is defined by
(2.2)[nk]q∶=[n]q![k]q![n-k]q!.
The q-derivative of a function f(x), denoted by Dqf, is defined by
(2.3)(Dqf)(x)∶=f(qx)-f(x)(q-1)x,x≠0,(Dqf)(0)∶=limx→0(Dqf)(x).
The formula for the q-derivative of a product and quotient are
(2.4)Dq(u(x)v(x))=Dq(u(x))v(x)+u(qx)Dq(v(x)).
Also, it is known that
(2.5)Dqxn=[n]qxn-1,DqE(ax)=aE(qax).
If |q|>1, or 0<|q|<1 and |z|<1/(1-q), the q-exponential function eq(x) was defined by Jackson
(2.6)eq(z)∶=∑k=0∞‍zk[k]q!.
If |q|>1, eq(z) is an entire function and
(2.7)eq(z)=∏j=0∞‍(1+(q-1)zqj+1),|q|>1.
There is another q-exponential function which is entire when 0<|q|<1 and which converges when |z|<1/|1-q| if |q|>1. To obtain it we must invert the base in (2.6), that is, q→1/q:
(2.8)Eq(z)∶=e1/q(z)=∑k=0∞‍qk(k-1)/2zk[k]q!.
We immediately obtain from (2.7) that
(2.9)Eq(z)=∏j=0∞‍(1+(1-q)zqj),0<|q|<1.
The q-difference equations corresponding to eq(z) and Eq(z) are
(2.10)Dqeq(az)=aeq(qz),DqEq(az)=aEq(qaz),D1/qeq(z)=D1/qE1/q(z)=E1/q(q-1z)=eq(q-1z),q≠0.
Let Cp be the set of all real valued functions f, continuous on [0,∞), such that wpf is uniformly continuous and bounded on [0,∞) endowed with the norm
(2.11)∥f∥p∶=supx∈[0,∞)wp(x)|f(x)|.
Here
(2.12)w0(x)∶=1,wp(x)∶=(1+xp)-1,ifp∈ℕ.
The corresponding Lipschitz classes are given for 0<α≤2 by
(2.13)Δh2f(x)∶=f(x+2h)-2f(x+h)+f(x),ωp2(f;δ)∶=sup0<h≤δ∥Δh2f∥p,Lipp2α∶={f∈Cp:ωp2(f;δ)=0(δα),δ→0+}.Now we introduce theq-parametric Szász-Mirakjan operator.Definition 2.1.
Letq>1 and n∈ℕ. For f:[0,∞)→R one defines the Szász-Mirakjan operator based on the q-integers
(2.14)Mn,q(f;x)∶=∑k=0∞‍f([k]q[n]q)1qk(k-1)/2[n]qkxk[k]q!eq(-[n]qq-kx).Similarly as a classical Szász-Mirakjan operatorSn, the operator Mn,q is linear and positive. Furthermore, in the case of q→1+ we obtain classical Szász-Mirakjan operators.MomentsMn,q(tm;x) are of particular importance in the theory of approximation by positive operators. From (2.14) one easily derives the following recurrence formula and explicit formulas for moments Mn,q(tm;x),  m=0,1,2,3,4.Lemma 2.2.
Letq>1. The following recurrence formula holds
(2.15)Mn,q(tm+1;x)=∑j=0m‍(mj)xqj[n]qm-jMn,q(tj;q-1x).Proof.
The recurrence formula (2.15) easily follows from the definition of Mn,q and q[k]q+1=[k+1]q as show below:
(2.16)Mn,q(tm+1;x)=∑k=0∞‍[k]qm+1[n]qm+11qk(k-1)/2[n]qkxk[k]q!eq(-[n]qq-kx)=∑k=1∞‍[k]qm[n]qm1qk(k-1)/2[n]qk-1xk[k-1]q!eq(-[n]qq-kx)=∑k=0∞‍(q[k]q+1)m[n]qm1qk(k+1)/2[n]qkxk+1[k]q!eq(-[n]qq-kq-1x)=∑k=0∞‍1[n]qm∑j=0m‍(mj)qj[k]qj1qk(k+1)/2[n]qkxk+1[k]q!eq(-[n]qq-kq-1x)=∑j=0m‍(mj)xqj[n]qm-j∑k=0∞‍[k]qj[n]qj1qk(k-1)/2[n]qkxk[k]q!qkeq(-[n]qq-kq-1x)=∑j=0m‍(mj)xqj[n]qm-jMn,q(tj;q-1x).Lemma 2.3.
The following identities hold for allq>1,  x∈[0,∞), n∈ℕ, and k≥0:
(2.17)xDqsnk(q;x)=[n]q([k]q[n]q-x)snk(q;x),Mn,q(tm+1;x)=x[n]qDqMn,q(tm;x)+xMn,q(tm;x),
where snk(q;x)∶=(1/qk(k-1)/2)([n]qkxk/[k]q!)eq(-[n]qq-kx).Proof.
The first identitiy follows from the following simple calculations(2.18)xDqsnk(q;x)=[k]q1qk(k-1)/2[n]qkxk[k]q!eq(-[n]q-kx)-xq-k[n]q1qk(k-1)/2[n]qkqkxk[k]q!eq(-[n]qq-kx)=[k]qsnk(q;x)-x[n]qsnk(q;x)=[n]q([k]q[n]q-x)snk(q;x).
The second one follows from the first:
(2.19)xDqMn,q(tm;x)=[n]q∑k=0∞‍([k]q[n]q)m([k]q[n]q-x)snk(q;x)=[n]q∑k=0∞‍([k]q[n]q)m+1snk(q;x)-[n]qx∑k=0∞‍([k]q[n]q)msnk(q;x)=[n]qMn,q(tm+1;x)-[n]qxMn,q(tm;x).Lemma 2.4.
Letq>1. One has
(2.20)Mn,q(1;x)=1,Mn,q(t;x)=x,Mn,q(t2;x)=x2+1[n]qx,Mn,q(t3;x)=x3+2+q[n]qx2+1[n]q2x,Mn,q(t4;x)=x4+(3+2q+q2)x3[n]q+(3+3q+q2)x2[n]q2+1[n]q3x.Proof.
For a fixedx∈R+, by the q-Taylor theorem [24], we obtain
(2.21)φn(t)=∑k=0∞‍(t-x)1/qk[k]1/q!D1/qkφn(x).
Choosing t=0 and taking into account
(2.22)(-x)1/qk=(-1)kxkq-k(k-1)/2,D1/qkeq(-[n]qx)=(-1)kq-k(k-1)/2[n]qkeq(-[n]qq-kx)
we get for φn(x)=eq(-[n]qx) that
(2.23)1=φn(0)=∑k=0∞‍(-1)kxkqk(k-1)/2[k]1/q!D1/qkφn(x)=∑k=0∞‍(-1)kxk[k]q!(-1)kq-k(k-1)/2[n]qkeq(-[n]qq-kx)=∑k=0∞‍[n]qkxk[k]q!qk(k-1)/2eq(-[n]qq-kz).
In other words Mn,q(1;x)=1.
Calculation ofMn,q(ti;x), i=1,2,3,4, based on the recurrence formula (2.17) (or (2.15)). We only calculate Mn,q(t3;x) and Mn,q(t4;x):
(2.24)Mn,q(t3;x)=x[n]qDqMn,q(t2;x)+xMn,q(t2;x)=x[n]q([2]qx+1[n]q)+x(x2+1[n]qx)=1[n]q2x+2+q[n]qx2+x3,Mn,q(t4;x)=x[n]qDqMn,q(t3;x)+xMn,q(t3;x)=x[n]q(1[n]q2+2+q[n]q[2]qx+[3]qx2)+x(1[n]q2x+2+q[n]qx2+x3)=1[n]q3x+(3+3q+q2)x2[n]q2+(3+2q+q2)x3[n]q+x4.Lemma 2.5.
Assume thatq>1. For every x∈[0,∞) there hold
(2.25)Mn,q((t-x)2;x)=x[n]q,(2.26)Mn,q((t-x)3;x)=1[n]q2x+(q-1)x2[n]q,(2.27)Mn,q((t-x)4;x)=1[n]q3x+(q2+3q-1)x2[n]q2+(q-1)2x3[n]q.Proof.
First of all we give an explicit formula forMn,q((t-x)4;x).
(2.28)Mn,q((t-x)3;x)=Mn,q(t3;x)-3xMn,q(t2;x)+3x2Mn,q(t;x)-x3=x3+2+q[n]qx2+1[n]q2x-3x(x2+x[n]q)+3x3-x3=1[n]q2x+(q-1)x2[n]q,Mn,q((t-x)4;x)=Mn,q(t4;x)-4xMn,q(t3;x)+6x2Mn,q(t2;x)-4x3Mn,q(t;x)+x4=1[n]q3x+(3+3q+q2)x2[n]q2+(3+2q+q2)x3[n]q+x4-4x(1[n]q2x+2+q[n]qx2+x3)+6x2(x2+x[n]q)-4x4+x4=1[n]q3x+(-1+3q+q2)x2[n]q2+(q-1)2x3[n]q.Now we prove explicit formula for the momentsMn,q(tm;x), which is a q-analogue of a result of Becker, see [26, Lemma 3].Lemma 2.6.
Forq>1,  m∈ℕ there holds
(2.29)Mn,q(tm;x)=∑j=1m‍𝕊q(m,j)xj[n]qm-j,
where
(2.30)𝕊q(m+1,j)=[j]𝕊q(m,j)+𝕊q(m,j-1),m≥0,j≥1,𝕊q(0,0)=1,𝕊q(m,0)=0,m>0,𝕊q(m,j)=0,m<j.
In particular Mn,q(tm;x) is a polynomial of degree m without a constant term.Proof.
Because ofMn,q(t;x)=x, Mn,q(t2;x)=x2+x/[n]q, the representation (2.29) holds true for m=1,2 with 𝕊q(2,1)=1,  𝕊q(1,1)=1.
Now assume (2.29) to be valued for m then by Lemma 2.3 we have
(2.31)Mn,q(tm+1;x)=x[n]qDqMn,q(tm;x)+xMn,q(tm;x)=x[n]q∑j=1m‍[j]q𝕊q(m,j)xj-1[n]qm-j+x∑j=1m‍𝕊q(m,j)xj[n]qm-j=∑j=1m‍[j]q𝕊q(m,j)xj[n]qm-j+1+∑j=1m‍𝕊q(m,j)xj+1[n]qm-j=x[n]m𝕊q(m,1)+xm+1𝕊q(m,m)+∑j=2m‍([j]q𝕊q(m,j)+𝕊q(m,j-1))xj[n]qm-j+1.Remark 2.7.
Notice that𝕊q(m,j) are Stirling numbers of the second kind introduced by Goodman et al. in [8]. For q=1 the formulae (2.30) become recurrence formulas satisfied by Stirling numbers of the second type.


## 3.Mn,q in Polynomial-Weighted Spaces

Lemma 3.1.
Letp∈ℕ∪{0} and q∈(1,∞) be fixed. Then there exists a positive constant K1(q,p) such that
(3.1)∥Mn,q(1/wp;x)∥p≤K1(q,p),n∈ℕ.
Moreover for every f∈Cp one has
(3.2)∥Mn,q(f)∥p≤K1(q,p)∥f∥p,n∈ℕ.
Thus Mn,q is a linear positive operator from Cp into Cp for any p∈ℕ∪{0}.Proof.
The inequality (3.1) is obvious for p=0. Let p≥1. Then by (2.29) we have
(3.3)wp(x)Mn,q(1/wp;x)=wp(x)+wp(x)∑j=1p‍𝕊q(p,j)xj[n]qp-j≤K1(q,p),K1(q,p) is a positive constant depending on p and q. From this follows (3.1). On the other hand
(3.4)∥Mn,q(f)∥p≤∥f∥p∥Mn,q(1wp)∥p,
for every f∈Cp. By applying (3.1), we obtain (3.2).Lemma 3.2.
Letp∈ℕ∪{0} and q∈(1,∞) be fixed. Then there exists a positive constant K2(q,p) such that
(3.5)∥Mn,q((t-·)2wp(t);·)∥p≤K2(q,p)[n]q,n∈ℕ.Proof.
The formula (2.25) imply (3.5) for p=0. We have
(3.6)Mn,q((t-x)2wp(t);x)=Mn,q((t-x)2;x)+Mn,q((t-x)2tp;x),
for p,n∈ℕ. If p=1 then we get
(3.7)Mn,q((t-x)2(1+t);x)=Mn,q((t-x)2;x)+Mn,q((t-x)2t;x)=Mn,q((t-x)3;x)+(1+x)Mn,q((t-x)2;x),
which by Lemma 2.5 yields (3.5) for p=1.
Letp≥2. By applying (2.29), we get
(3.8)wp(x)Mn,q((t-x)2tp;x)=wp(x)(Mn,q(tp+2;x)-2xMn,q(tp+1;x)+x2Mn,q(tp;x))=wp(x)(xp+2+∑j=1p+1‍𝕊q(p+2,j)xj[n]qp+2-j-2xp+2-2∑j=1p‍𝕊q(p+1,j)xj+1[n]qp+1-j+xp+2+∑j=1p-1‍𝕊q(p,j)xj+2[n]qp-j)=wp(x)(∑j=2p‍(𝕊q(p+2,j)-2𝕊q(p+1,j)+𝕊q(p,j))xj+1[n]qp+1-j+𝕊q(p+2,1)x[n]qp+1+(𝕊q(p+2,2)-2𝕊q(p+2,1))x2[n]qp)=wp(x)x[n]q𝒫p(q;x),
where 𝒫p(q;x) is a polynomial of degree p. Therefore one has
(3.9)wp(x)Mn,q((t-x)2tp;x)≤K2(q,p)x[n]q.Our first main result in this section is a local approximation property ofMn,q stated below.Theorem 3.3.
There exists an absolute constantC>0 such that
(3.10)wp(x)|Mn,q(g;x)-g(x)|≤K3(q,p)∥g′′∥x[n]q,
where g∈Cp2, q>1 and x∈[0,∞).Proof.
Using the Taylor formula(3.11)g(t)=g(x)+g′(x)(t-x)+∫xt‍∫xs‍g′′(u)duds,g∈Cp2,
we obtain that
(3.12)wp(x)|Mn,q(g;x)-g(x)|=wp(x)|Mn,q(∫xt‍∫xs‍g′′(u)duds;x)|≤wp(x)Mn,q(|∫xt‍∫xs‍g′′(u)duds|;x)≤wp(x)Mn,q(∥g′′∥p|∫xt‍∫xs‍(1+um)duds|;x)≤wp(x)12∥g′′∥pMn,q((t-x)2(1/wp(x)+1/wp(t));x)≤12∥g′′∥p(Mn,q((t-x)2;x)+wp(x)Mn,q((t-x)2wp(t);x))≤K3(q,x)∥g′′∥px[n]q.Now we consider the modified Steklov means(3.13)fh(x)∶=4h2∬0h/2[2f(x+s+t)-f(x+2(s+t))]dsdt.fh(x) has the following properties:
(3.14)f(x)-fh(x)=4h2∬0h/2Δs+t2f(x)dsdt,fh′′(x)=h-2(8Δh/22f(x)-Δh2f(x))
and therefore
(3.15)∥f-fh∥p≤ωp2(f;h),∥fh′′∥p≤19h2ωp2(f;h).
We have the following direct approximation theorem.Theorem 3.4.
For everyp∈ℕ∪{0},f∈Cp and x∈[0,∞),  q>1, one has
(3.16)wp(x)|Mn,q(f;x)-f(x)|≤Mpωp2(f;x[n]q)=Mpωp2(f;(q-1)x(qn-1)).
Particularly, if Lipp2α for some α∈(0,2], then
(3.17)wp(x)|Mn,q(f;x)-f(x)|≤Mp(x[n]q)α/2.Proof.
Forf∈Cp and h>0(3.18)|Mn,q(f;x)-f(x)|≤|Mn,q((f-fh);x)-(f-fh)(x)|+|Mn,q(fh;x)-fh(x)|
and therefore
(3.19)wp(x)|Mn,q(f;x)-f(x)|≤∥f-fh∥p(wp(x)Mn,q(1wp(t);x)+1)+K3(q,p)∥fh′′∥px[n]q.
Since wp(x)Mn,q(1/wp(t);x)≤K1(q,p), we get that
(3.20)wp(x)|Mn,q(f;x)-f(x)|≤M(q,p)ωp2(f;h)[1+x[n]qh2].
Thus, choosing h=x/[n]q, the proof is completed.Corollary 3.5.
Ifp∈ℕ∪{0},f∈Cp,  q>1 and x∈[0,∞),  then
(3.21)limn→∞Mn,q(f;x)=f(x).
This converegnce is uniform on every [a,b], 0≤a<b.Remark 3.6.
Theorem3.4 shows the rate of approximation by the q-Szász-Mirakjan operators (q>1) is of order q-n versus 1/n for the classical Szász-Mirakjan operators.


## 4. Convergence ofq-Szász-Mirakjan Operators

An interesting problem is to determine the class of all continuous functionsf such that Mn,q(f) converges to f uniformly on the whole interval [0,∞) as n→∞. This problem was investigated by Totik [27, Theorem 1] and de la Cal and Cárcamo [28, Theorem 1]. The following result is a q-analogue of Theorem 1 [28].Theorem 4.1.
Assume thatf:[0,∞)→R is bounded or uniformly continuous. Let
(4.1)f*(z)=f(z2),z∈[0,∞).
One has, for all t>0 and x≥0,
(4.2)|Mn,q(f;x)-f(x)|≤2ω(f*;1[n]q).
Therefore, Mn,q(f;x) converges to f uniformly on [0,∞) as n→∞, whenever f* is uniformly continuous.Proof.
By the definition off* we have
(4.3)Mn,q(f;x)=Mn,q(f*(·);x).
Thus we can write
(4.4)|Mn,q(f;x)-f(x)|=|Mn,q(f*(·);x)-f*(x)|=|∑k=0∞‍(f*([k]q[n]q)-f*(x))sn,k(q;x)|≤∑k=0∞‍|(f*([k]q[n]q)-f*(x))|sn,k(q;x)≤∑k=0∞‍ω(f*;|[k]q[n]q-x|)sn,k(q;x)≤∑k=0∞‍ω(f*;|[k]q/[n]q-x|Mn,q(|·-x|;x)Mn,q(|·-x|;x))sn,k(q;x).
Finally, from the inequality
(4.5)ω(f*;αδ)≤(1+α)ω(f*;δ),α,δ≥0,
we obtain
(4.6)|Mn,q(f;x)-f(x)|≤ω(f*;Mn,q(|·-x|;x))∑k=0∞‍(1+|[k]q/[n]q-x|Mn,q(|·-x|;x))sn,k(q;x)=2ω(f*;Mn,q(|·-x|;x)).
In order to complete the proof we need to show that we have for all t>0 and x>0,
(4.7)Mn,q(|·-x|;x)≤1[n]q.
Indeed we obtain from the Cauchy-Schwarz inequality:
(4.8)Mn,q(|·-x|;x)=∑k=0∞‍|[k]q[n]q-x|sn,k(q;x)=∑k=0∞‍|[k]q/[n]q-x|[k]q/[n]q+xsn,k(q;x)≤1x∑k=0∞‍|[k]q[n]q-x|sn,k(q;x)≤1x∑k=0∞‍|[k]q[n]q-x|2sn,k(q;x)=1xMn,q((·-x)2;x)=1x1[n]qx=1[n]q
showing (4.2), and completing the proof.Next we prove Voronovskaja type result forq-Szász-Mirakjan operators.Theorem 4.2.
Assume thatq∈(1,∞). For any f∈Cp2 the following equality holds
(4.9)limn→∞[n]q(Mn,q(f;x)-f(x))=12f′′(x)x,
for every x∈[0,∞).Proof.
Letx∈[0,∞) be fixed. By the Taylor formula we may write
(4.10)f(t)=f(x)+f′(x)(t-x)+12f′′(x)(t-x)2+r(t;x)(t-x)2,
where r(t;x) is the Peano form of the remainder, r(·;x)∈Cp, and limt→xr(t;x)=0. Applying Mn,q to (4.10) we obtain
(4.11)[n](Mn,q(f;x)-f(x))=f′(x)[n]qMn,q(t-x;x)+12f′′(x)[n]qMn,q((t-x)2;x)+[n]qMn,q(r(t;x)(t-x)2;x).
By the Cauchy-Schwartz inequality, we have
(4.12)Mn,q(r(t;x)(t-x)2;x)≤Mn,q(r2(t;x);x)Mn,q((t-x)4;x).
Observe that r2(x;x)=0. Then it follows from Corollary 3.5 that
(4.13)limn→∞Mn,q(r2(t;x);x)=r2(x;x)=0.
Now from (4.12), (4.13), and Lemma 2.5 we get immediately
(4.14)limn→∞[n]qMn,q(r(t;x)(t-x)2;x)=0.
The proof is completed.



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