1. Introduction
Let q>0. For each nonnegative integer k, the q-integer [k] and the q-factorial [k]! are defined by
(1)[k]:=[k]q:={1-qk1-q,q≠1,k,q=1,[k]!:={[k][k-1]⋯[1],k≥1,1,k=0.
For integers 0≤k≤n, the q-binomial coefficient is defined by
(2)[nk]:=[n]![k]![n-k]!.
We give the following two q-analogues of exponential function ex:
(3)eq(x):=∑k=0∞xk[k]!=1((1-q)x;q)∞,|x|<11-q for q<1;Eq(x):=∑k=0∞qk(k-1)/2xk[k]!=(-(1-q)x;q)∞,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiix∈ℝ for q<1,
where (x;q)∞:=∏k=1∞(1-xqk-1). Clearly, we have
(4)eq(x)Eq(-x)=1, limq→1-eq(x)=limq→1-Eq(x)=ex.
In [1], Phillips proposed the q-Bernstein polynomials: for each positive integer n and f∈C[0,1], the q-Bernstein polynomial of f is
(5)Bn,q(f)(x):=∑k=0nf([k][n])[nk]xk∏s=0n-k-1(1-qsx).
Note that for q=1, Bn,q(f) is the classical Bernstein polynomial. In [2], II’inskiia and Ostrovska proved that, for each f∈C[0,1] and q∈(0,1), the sequence {Bn,q(f)(x)} converges to B∞,q(f)(x) as n→∞ uniformly on x∈[0,1], where
(6)B∞,q(f)(x)={∑k=0∞f(1-qk)p∞k(q;x),0≤x<1,f(1),x=1.
The operators B∞,q are called the limit q-Bernstein operators. They also arise as the limit for a sequence of q-Meyer-König Zeller operators (see [3]). For results about properties of B∞,q(f,x) we refer to [2, 4, 5].
In [6], Aral introduced the following q-Szász-Mirakjan operator: for each positive integer n and f∈C[0,∞),
(7)Sn,qb(f)(x):=Eq(-[n]xbn)∑k=0∞f([k]bn[n])([n]x)k[k]!(bn)k,
where 0≤x<αq(n), αq(n):=bn/(1-qn), and b={bn} is a sequence of positive numbers such that limn→∞bn=∞. In this paper, we introduce the following q-Szász-Mirakjan operator: for each positive integer n and f∈C[0,1/(1-qn)],
(8)Sn,q(f)(x) :={Eq(-[n]x)∑k=0∞f([k][n])([n]x)k[k]!,x∈[0,11-qn),f(1),x=11-qn.
Obviously, the operators Sn,q are equal to the operators Sn,qb with b={bn}, bn=1. When q=1, the q-Szász-Mirakjan operators Sn,q reduce to the classical Szász-Mirakjan operators.
In recent years, generalizations of linear operators connected with q-Calculus have been investigated intensively. The pioneer work has been made by Lupas [7] and Phillips [1] who proposed generalizations of Bernstein polynomials based on the q-integers. There are also other important q-operators, for example, the two-parametric generalization of q-Bernstein polynomials [8], the q-Bernstein-Durrmeyer operator [9], q-Meyer-König Zeller operators [10], q-Bleimann, Butzer and Hahn operators [11], and q-Szász-Mirakjan operators [6, 12–15]. Among these generalizations, q-Bernstein polynomials proposed by Phillips attracted the most attention and were studied widely by a number of authors (see [1, 2, 5, 16–24]).
In this paper, we will discuss convergence and shape-preserving properties of the q-Szász-Mirakjan operators Sn,q for fixed q∈(0,1). We will show that the operators Sn,q share good shape-preserving properties such as the variation-diminishing properties, and for each f∈C[0,1/(1-q)] the sequence {Sn,q(f)(x)} converges to the function B∞,q(f)(x) uniformly on [0,1], where B∞,q are the limit q-Bernstein operators defined by (6). We also investigate the rate of convergence of the q-Szász-Mirakjan operators Sn,q for fixed q∈(0,1). Our results demonstrate that in general convergence properties of the q-Szász-Mirakjan operators Sn,q are essentially different from those for the classical Szász-Mirakjan operators; however, they are very similar to those for the q-Bernstein polynomials. Notice that different q-generalizations of Szász-Mirakjan operators were introduced and studied by Aral and Gupta [6, 12], by Radu [13], and by Mahmudov [14, 15]. However, our q-Szász-Mirakjan operators have better convergence properties than the other q-generalizations of Szász-Mirakjan operators for fixed q∈(0,1).
The paper is organized as follows. In Section 2, we recall some properties of the q-Szász-Mirakjan operators Sn,q and discuss their shape-preserving properties. In Section 3 we investigate the convergence of Sn,q(f) for fixed q∈(0,1) and obtain the rate of convergence of Sn,q(f) by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.
2. Shape-Preserving Properties of Sn,q for 0<q<1
In the sequel we always assume that q∈(0,1). First we show that the q-Szász-Mirakjan operators Sn,q are the positive linear operators on C[0,1/(1-qn)]. Clearly, it suffices to prove that, for f∈C[0,1/(1-qn)],
(9)limx→(1/(1-qn))-Sn,q(f)(x)=f(11-qn).
Indeed, for arbitrary ɛ>0, there exist a constant M>0 and a δ>0 such that |f(x)|≤M for all x∈[0,1/(1-qn)], and |f(x)-f(1/(1-qn))|≤ɛ for x∈(1/(1-qn)-δ,1/(1-qn)). We choose A to be the minimum positive integer greater than logq((1-qn)δ. Then, for any k>A,
(10)|[k][n]-11-qn|=qk1-qn<δ,|f([k][n])-f(11-qn)|≤ɛ.
It follows from the Euler identity that
(11)Eq(-[n]x)∑k=0∞([n]x)k[k]!=1 for x∈[0,11-qn),Eq(-[n]x)=((1-qn)x;q)∞=∏s=0∞(1-qs(1-qn)x)⟶0+,iiiiiiiiiiiiiiiiiiiiias x⟶11-qn-.
This implies that, for x∈[0,1/(1-qn)),
(12)|Sn,q(f)(x)-f(11-qn)| =|Eq(-[n]x)∑k=0∞(f([k][n])-f(11-qn))([n]x)k[k]!| ≤Eq(-[n]x)(∑k=0A|f([k][n])-f(11-qn)|([n]x)k[k]! +∑k=A+1∞|f([k][n])-f(11-qn)|([n]x)k[k]!) ≤2MEq(-[n]x)∑k=0A1(1-q)k[k]! +ɛEq(-[n]x)∑k=A+1∞([n]x)k[k]! ≤BEq(-[n]x)+ɛ,
where B:=2M∑k=0A(1/(1-q)k[k]!) is a constant independent of x and Eq(-[n]x)→0 as x→(1/(1-qn))-. This proves (9).
The q-Szász-Mirakjan operators Sn,q possess the end-point interpolation property:
(13)Sn,q(f)(0)=f(0), Sn,q(f)(11-qn)=f(11-qn),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiin∈ℕ.
They leave invariant linear functions:
(14)Sn,q(at+b)(x)=ax+b
and are degree-preserving on polynomials; that is, if T is a polynomial of degree m, then Sn,q(T) is a polynomial of degree m (see [6, Lemma 1] or [25, Theorem 1]).
The following representation of the q-Szász-Mirakjan operators Sn,q, called the q-difference form, was obtained in [6, Corollary 4]:
(15)Sn,q(f)(x)=∑k=0∞qk(k-1)/2f([[0][n];[1][n];…;[k][n]])xk,
where f([x0;x1;…;xk]) denotes the usual divided difference; that is,
(16)f([x0])=f(x0); f([x0;x1])=f(x1)-f(x0)x1-x0,…,f([x0;x1;…;xk])=f([x1;…;xk])-f([x0;…;xk-1])xk-x0.
Aral and Gupta discussed the shape-preserving properties of the q-Szász-Mirakjan operators in [12, Corollary 3.2]. We say a function f on an interval I is i-convex, i≥1, if f∈C(I) and all ith forward differences
(17)Δhif(t):=∑k=0i(-1)i-k(ik)f(t+kh),iiiiiiiiiiiiiiiiiii0≤h≤1i, t,t+kh∈I
are nonnegative. Obviously, a 1-convex function is nondecreasing and a 2-convex function is convex. Aral and Gupta obtained that, for an i-convex function on [0,∞), there exists q^∈(0,1) such that Sn,q(f) is also i-convex on [0,1/(1-qn)) for q∈(q^,1).
In this section we also study the shape-preserving properties of the operators Sn,q. We use a completely different method from the one in [12], and our results hold for all q∈(0,1). In order to state the results, we introduce some notations.
For any real sequence a, finite or infinite, we denote by S-(a) the number of strict sign changes in a. For f∈C(I), where I is an interval, we define S-(f) to be the number of sign changes of f; that is,
(18)S-(f)=sup S-(f(x0),…,f(xm)),
where the supremum is taken over all increasing sequences x0<⋯<xm and x0,xm∈I for all positive integers m.
Let L be a positive linear operator on C(I). We say that L is variation-diminishing if, for all functions f∈C(I), we have
(19)S-(Lnf)≤S-(f).
A function ω(t) on [0,A], A>0 is called a modulus of continuity if ω(t) is continuous, nondecreasing, and semiadditive and ω(0)=0. We denote by Hω the class of continuous functions f on [0,A] satisfying the inequality ω(f,t)≤ω(t), where ω(f,t)=max|x1-x2|≤t|f(x2)-f(x1)| is the modulus of continuity of f(x). Note that if f(x) is a concave modulus of continuity, then x-1f(x) is nonincreasing on (0,A]. Also, if f(x) is a nondecreasing function such that f(0)=0 and x-1f(x) is nonincreasing on (0,A], then f(x) is a modulus of continuity.
Our main results of this section can be formulated as follows.
Theorem 1.
(i) The operators Sn,q are variation-diminishing on [0,1/(1-qn)].
(ii) If a function f is i-convex on [0,1/(1-qn)], then the functions Sn,q(f) are also i-convex on [0,1/(1-qn)]. Specially, if a function f is nondecreasing (nonincreasing) on [0,1/(1-qn)], then Sn,q(f) are also nondecreasing (nonincreasing) on [0,1/(1-qn)] and if f is convex (concave) on [0,1/(1-qn)], then so are Sn,q(f).
(iii) If a function f is convex on [0,1/(1-qn)], then Sn,q(f)(x)≥f(x), x∈[0,1/(1-qn)].
(iv) If ω(t) is a modulus of continuity, then f∈Hω implies that, for each n≥1, Sn,q(f)∈H2ω; if ω(t) is concave, then, for each n≥1, Sn,q(f)∈Hω.
(v) If ω(t) is a concave modulus of continuity, then, for each n≥1, Sn,q(ω) is also a concave modulus of continuity and Sn,q(ω)(t)≤ω(t).
(vi) If f(x) is a nonnegative function such that x-1f(x) is nonincreasing on (0,1/(1-qn)], then, for each n≥1, x-1Sn,q(f)(x) is nonincreasing also.
Proof.
(i) Let I be an interval, I⊂[0,∞). We assume that, for a real sequence a={ak}k=0∞, the power series ∑k=0∞akxk converges to the function g on I. By means of the well-known Descartes’ rule of sign it is easy to prove that
(20)S-(g)=S-(∑k=0∞akxk)≤S-(a).
Obviously, if h(x)>0 for any x∈I and bk>0 for k≥0, then
(21)S-(f)=S-(f·h), S-({akbk}k=0∞)=S-({ak}k=0∞).
It follows that
(22)S-(Sn,q(f))=S-(∑k=0∞f([k][n])[n]k[k]!xk)≤S-({f([k][n])}k=0∞)≤S-(f),
which implies that Sn,q are variation-diminishing.
(ii) The operators Sn,q possess the end-point interpolation property and are degree-preserving on polynomials and variation-diminishing. Then, (ii) follows from [26, Lemma 15].
(iii) It follows from [27, p. 281] that if a positive operator L on C[0,A] reproduces linear functions, then L(f,x)≥f(x) for any convex function f and for any x∈[0,A]. Since Sn,q are the positive linear operators and reproduce linear functions, we obtain (iii).
(iv) From [26, Corollary 8], we know that if a positive linear operator L on C[0,A] (A>0) is variation-diminishing and reproduces linear functions, then, for all f∈C[0,A] and t∈(0,A],
(23)ω(Lf,t)≤ω~(f,t).
Thus, if f∈Hω, then
(24)ω(Sn,q(f),t)≤ω~(f,t)≤ω~(t),
where ω~(t) and ω~(f,t) denote the least concave majorant of ω(t) and ω(f,t), respectively. It is well known that for each modulus of continuity ω there exists a concave modulus of continuity ω~ such that ω(t)≤ω~(t)≤2ω(t) for t∈[0,A]. Thence, Sn,q(f)∈H2ω and furthermore Sn,q(f)∈Hω if ω is concave, which means (iv) holds.
(v) From (i) we know that, for a concave modulus of continuity ω and each n≥1, the function Sn,q(ω) is nondecreasing and concave on (0,A], where A=1/(1-qn). We also have Sn,q(ω)(0)=0. This means that Sn,q(ω) is a concave modulus of continuity. The inequality Sn,q(ω)(t)≤ω(t) follows directly from (iii).
(vi) Since, for any constant c,
(25)S-(Sn,q(f)(x)x-c) =S-(Sn,q(f)(x)-cx)=S-(Sn,q(f(t)-ct)(x)) ≤S-(f(x)-cx)=S-(f(x)x-c)≤1
we get that Sn,q(f)(x)/x is nondecreasing or nonincreasing on (0,A], where A=1/(1-qn). For any t∈(0,A), f(t)/t≥f(A)/A, we have f(t)≥f(A)t/A, and thus Sn,q(f)(x)≥Sn,q(f(A)t/A)(x)=f(A)x/A. Hence,
(26)Sn,q(f)(x)x≥f(A)A=Sn,q(f)(A)A,
which implies that Sn,q(f)(x)/x is nonincreasing on [0,A].
Theorem 1 is proved.
3. The Rate of Convergence for the q-Szász-Mirakjan Operators Sn,q for Fixed q∈(0,1)
The approximation properties of the sequence {Sn,qnb(f)} in weighted spaces as limn→∞qn=1- were investigated in [6, Theorem 2] and [25, Theorem 6]. The obtained results are similar to the ones of the classical Szász-Mirakjan operators. However, there are few results about convergence properties of Sn,q for fixed q∈(0,1). This section is devoted to discussing the convergence properties of the q-Szász-Mirakjan operators Sn,q for fixed q∈(0,1).
We set
(27)sn,k(q;x)=Eq(-[n]x)([n]x)k[k]!=([n]x)k[k]!((1-qn)x;q)∞,(28)p∞,k(q;x)=xk(1-q)k[k]!(x;q)∞.
Formerly, for f∈C[0,1/(1-q)] and each k≥0, {f([k]/[n])} converges to f(1-qk), {sn,k(q;x)} converges to p∞,k(q;x), and
(29)Sn,q(f)(x)=∑k=0∞f([k][n])sn,k(q;x)⟶∑k=0∞f(1-qk)p∞,k(q;x)=B∞,q(f)(x),
as n→∞. Indeed, the above conclusion holds. We have the following stronger results.
Theorem 2.
Let f∈C[0,1/(1-q)]. Then, we have
(30)supx∈[0,1]|Sn,q(f)(x)-B∞,q(f)(x)|≤Cqω(f,qn),
where Cq=4+q/(1-q)+(1/(1-q))eq2/(1-q)2. This estimate is sharp in the following sense of order: for each α, 0<α≤1, there exists a function fα(x) which belongs to the Lipschitz class Lipα∶={f∈C[0,1]∣ω(f;t)≤tα} such that
(31)supx∈[0,1]|Sn,q(fα)(x)-B∞,q(fα)(x)|≥Cqnα,
where C is a positive constant independent of n.
Remark 3.
It follows from (30) that, for f∈C[0,1/(1-q)], limn→∞Sn,q(f)(x)=B∞,q(f)(x) uniformly on x∈[0,1] as n→∞. Since B∞,q(f)(x)=f(x), x∈[0,1], if and only if f is linear on [0,1] (see [2, Theorem 6]), we get that the sequence Sn,q(f)(x) converges to f uniformly on [0,1] if and only if f is linear on [0,1].
Remark 4.
It should be emphasized that the proof of Theorem 2 requires estimation techniques involving the infinite product. Also, it is a little more difficult than the one used for q-Bernstein polynomials (see [23]), since Sn,q(f)(1)≠f(1)=B∞,q(f)(1).
Proof.
Since the operators Sn,q and B∞,q reproduce linear functions, we get that, for x∈[0,1),
(32)∑k=0∞sn,k(q;x)=1, ∑k=0∞[k][n]sn,k(q;x)=x,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiix∈[0,11-qn),(33)∑k=0∞p∞,k(q;x)=1, ∑k=0∞(1-qk)p∞,k(q;x)=x,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiix∈[0,1),
where sn,k(q;x) and p∞,k(q;x) are defined by (27) and (28), respectively. By means of (32) and (33), direct calculations give that
(34)∑k=0nqksn,k(q;x)=1-x+qnx, ∑k=0∞qkp∞,k(q;x)=1-x.
For x=0, we have
(35)|Sn,q(f)(0)-B∞,q(f)(0)|=|f(0)-f(0)|=0.
For x=1, it follows that
(36)|Sn,q(f)(1)-B∞,q(f)(1)| =|∑k=0∞(f([k][n])-f(1))sn,k(q;1)| ≤∑k=0∞(|f([k][n])-f(1-qk)| +|f(1)-f(1-qk)||f([k][n])-f(1-qk)|)sn,k(q;1) ≤∑k=0∞(ω(f,[k]qn[n])+ω(f,qk))sn,k(q;1) ≤ω(f,qn)∑k=0∞(2+[k][n]+qkqn)sn,k(q;1) =4ω(f,qn),
where in the first equality we used (32); in the last inequality we used the inequality ω(f,λt)≤(1+λ)ω(f,t) for any λ,t>0; in the last equality we used (32) and (34).
Now for x∈(0,1), by (32) and (33), we have
(37)|Sn,q(f,x)-B∞,q(f,x)| =|∑k=0∞f([k][n])sn,k(q;x)-∑k=0∞f(1-qk)p∞,k(q;x)| =|∑k=0∞(f([k][n])-f(1-qk))sn,k(q;x) +∑k=0∞(f(1-qk)-f(1))(sn,k(q;x)-p∞,k(q;x))| ≤∑k=0∞|f([k][n])-f(1-qk)|sn,k(q;x) +∑k=0∞|f(1-qk)-f(1)||sn,k(q;x)-p∞,k(q;x)| =:J1+J2.
Since
(38)|[k][n]-(1-qk)|=[k]qn[n],ω(f;λt)≤(1+λ)ω(f;t), λ,t>0,
we get by (32)
(39)J1≤∑k=0∞ω(f,[k]qn[n])sn,k(q;x)≤ω(f,qn)∑k=0∞(1+[k][n])sn,k(q;x)=(1+x)ω(f,qn)≤2ω(f,qn).
In order to estimate J2, we need to estimate |sn,k(q;x)-p∞,k(q;x)|. We have
(40)|sn,k(q;x)-p∞,k(q;x)| =|([n]x)k[k]!∏s=0∞(1-(1-qn)qsx) -xk(1-q)k[k]!∏s=0∞(1-qsx)| ≤xk(1-q)k[k]!|∏s=0∞(1-qs(1-qn)x)(1-qn)k -∏s=0∞(1-qsx)(1-qn)k| +xk(1-q)k[k]!∏s=0∞(1-qsx)|1-(1-qn)k| ≤p∞,k(q;x)(|∏s=0∞(1+qs+nx1-qsx)-1|+|1-(1-qn)k|).
We note that
(41)qk(1-(1-qn)k)=qk+n(1+(1-qn)+⋯+(1-qn)k-1)≤kqk+n≤qn+11-q.
It follows that
(42)J2≤∑k=0∞ω(f,qk)|sn,k(q;x)-p∞,k(q;x)|≤ω(f,qn)∑k=0∞(1+qkqn)|sn,k(q;x)-p∞,k(q;x)|≤ω(f,qn)(∑k=0∞(sn,k(q;x)+p∞,k(q;x)) +∑k=0∞qkqn|sn,k(q;x)-p∞,k(q;x)|)≤ω(f,qn)(2+∑k=0∞qkqnp∞,k(q;x) ×(∑k=0∞|∏s=0∞(1+qs+nx1-qsx)-1| +|1-(1-qn)k|∑k=0∞))≤ω(f,qn)(2+∑k=0∞p∞,k(q;x) ×qk-n|∏s=0∞(1+qs+nx1-qsx)-1|+q1-q)∑k=0∞)≤ω(f,qn)(|∏s=0∞(1+qs+nx1-qsx)-1|2+q-n(1-x) ×|∏s=0∞(1+qs+nx1-qsx)-1|+q1-q)=:ω(f,qn)(2+H+q1-q),
where in the fourth inequality we used (32) and (33); in the last inequality we used (34) and (33). We estimate H. We have
(43)H=q-n|(1-x+qnx)∏s=1∞(1+qs+nx1-qsx)-(1-x)|=x∏s=1∞(1+qs+nx1-qsx)+q-n(1-x) ×|∏s=1∞(1+qs+nx1-qsx)-1|=:xeK+q-n(1-x)(eK-1),
where K:=∑s=1∞ln(1+qs+nx/(1-qsx)). Using the inequality ln(1+t)≤t,t≥0, we get that
(44)K≤∑s=1∞qs+nx1-qsx≤∑s=1∞qs+n1-qx≤qn+1(1-q)(1-qx)≤q2(1-q)2.
It follows that
(45)eK≤eq2/(1-q)2,eK-1=Keξ≤KeK≤qn+1(1-q)(1-qx)eq2/(1-q)2,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiξ∈[0,K].
This deduces that, for x∈(0,1),
(46)H≤eq2/(1-q)2+(1-x)q(1-q)(1-qx)eq2/(1-q)2≤11-qeq2/(1-q)2,
and thence
(47)J2≤ω(f,qn)(2+q1-q+11-qeq2/(1-q)2).
We conclude from (39) and (47) that, for x∈(0,1),
(48)|Sn,q(f,x)-B∞,q(f,x)|≤J1+J2 ≤ω(f,qn)(4+q1-q+11-qeq2/(1-q)2).
Hence, (30) follows from (35), (36), and (48).
At last we show that the estimate (30) is sharp. For each α, 0<α≤1, suppose that fα*(x) is a continuous function, which is equal to zero in [0,1-q] and [1-q2,1], equal to (x-(1-q))α in [1-q,1-q+q(1-q)/2], and linear in the rest of [0,1]. It is easy to see that ω(fα*,t)≤Atα. We set fα(t)=(1/A)fα*(t). Then, fα∈Lipα, and for sufficiently large n, we have
(49)supx∈[0,1]|Sn,q(fα)(x)-B∞,q(fα)(x)| =1A(1-q)αqnα(1-qn)αsupx∈[0,1]|sn,1(q;x)| ≥(1-q)αqnαA|sn,1(q;12)| ≥(1-q)α2A(1-q)∏s=0∞(1-qs2)qnα=:Cqnα.
The proof of Theorem 2 is complete.