# Shape-Preserving and Convergence Properties for theq-Szász-Mirakjan Operators for Fixed q∈(0,1)

Authors: Heping Wang; Fagui Pu; Kai Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We introduce aq-generalization of Szász-Mirakjan operators Sn,q and discuss their properties for fixed q∈(0,1). We show that the q-Szász-Mirakjan operators Sn,q have good shape-preserving properties. For example, Sn,q are variation-diminishing, and preserve monotonicity, convexity, and concave modulus of continuity. For fixed q∈(0,1), we prove that the sequence {Sn,qf} converges to B∞,q(f) uniformly on [0,1] for each f∈C[0,1/(1-q)], where B∞,q is the limit q-Bernstein operator. We obtain the estimates for the rate of convergence for {Sn,qf} by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.


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## Body

## 1. Introduction

Letq>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-qforq<1;Eq(x):=∑k=0∞‍qk(k-1)/2xk[k]!=(-(1-q)x;q)∞,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiix∈ℝforq<1,
where (x;q)∞:=∏k=1∞‍(1-xqk-1). Clearly, we have
(4)eq(x)Eq(-x)=1,lim⁡q→1-eq(x)=lim⁡q→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 lim⁡n→∞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 withq-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 theq-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 Section2, 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 ofSn,q for 0<q<1

In the sequel we always assume thatq∈(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)lim⁡x→(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 log⁡q((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]!=1forx∈[0,11-qn),Eq(-[n]x)=((1-qn)x;q)∞=∏s=0∞‍(1-qs(1-qn)x)⟶0+,iiiiiiiiiiiiiiiiiiiiiasx⟶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).Theq-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 theq-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 theq-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 operatorsSn,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 sequencea, 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.LetL 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 functionω(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 operatorsSn,q are variation-diminishing on [0,1/(1-qn)].
(ii) If a functionf 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 functionf is convex on [0,1/(1-qn)], then Sn,q(f)(x)≥f(x), x∈[0,1/(1-qn)].
(iv) Ifω(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) Ifω(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) Iff(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) LetI 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 operatorsSn,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 continuityω 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 constantc,
(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].
Theorem1 is proved.


## 3. The Rate of Convergence for theq-Szász-Mirakjan Operators Sn,q for Fixed q∈(0,1)

The approximation properties of the sequence{Sn,qnb(f)} in weighted spaces as lim⁡n→∞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.
Letf∈C[0,1/(1-q)]. Then, we have
(30)sup⁡x∈[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 Lip⁡α∶={f∈C[0,1]∣ω(f;t)≤tα} such that
(31)sup⁡x∈[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)], lim⁡n→∞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 Theorem2 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 operatorsSn,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 forx∈(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∈Lip⁡α, and for sufficiently large n, we have
(49)sup⁡x∈[0,1]|Sn,q(fα)(x)-B∞,q(fα)(x)|=1A(1-q)αqnα(1-qn)αsup⁡x∈[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.



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*Source: 563613-2014-05-06.xml*