# A Survey of Results  on the Limitq-Bernstein Operator

Authors: Sofiya Ostrovska
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The limitq-Bernstein operator Bq emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution, which is used in the q-boson theory to describe the energy distribution in a q-analogue of the coherent state. At the same time, this operator bears a significant role in the approximation theory as an exemplary model for the study of the convergence of the q-operators. Over the past years, the limit q-Bernstein operator has been studied widely from different perspectives. It has been shown that Bq is a positive shape-preserving linear operator on C[0,1] with ∥Bq∥=1. Its approximation properties, probabilistic interpretation, the behavior of iterates, and the impact on the smoothness of a function have already been examined. In this paper, we present a review of the results on the limit q-Bernstein operator related to the approximation theory. A complete bibliography is supplied.


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## Body

## 1. Introduction

The limitq-Bernstein operator comes out as an analogue of the Szász-Mirakyan operator related to the Euler probability distribution, also called the q-deformed Poisson distribution (see [1–3]). The latter is used in the q-boson theory, which is a q-deformation of the quantum harmonic oscillator formalism [4]. Namely, the q-deformed Poisson distribution describes the energy distribution in a q-analogue of the coherent state [5]. The q-analogue of the boson operator calculus has proved to be a powerful tool in theoretical physics, providing explicit expressions for the representations of the quantum group SUq(2), which itself is by now known to play a profound role in a variety of different problems. Some of these are integrable model field theories, exactly solvable lattice models of statistical mechanics, conformal field theory, and others. Therefore, the properties of the q-deformed Poisson distribution and its related limit q-Bernstein operator have proved to be of paramount value for various applications. What is more, this operator is also decisive for the approximation theory as a model pertinent to the asymptotic behavior for a sequence of the q-operators. Indeed, operators whose nature is similar to that of Bq appear as a limit of a sequence of the various q-operators, see, for example, [6–11]. In this respect, a general approach has been developed by Wang in [12].To present the subject of this survey, it can serve well to recall some notions related to theq-calculus (cf., e.g., [13]).Letq>0. For any k∈ℤ+, the q-integer[k]q is defined by
(1)[k]q≔1+q+⋯+qk-1(k∈ℕ),[0]q≔0,
and the q-factorial[k]q! by
(2)[k]q!≔[1]q[2]q⋯[k]q(k=1,2,…),[0]q!≔1.
For integers k and n with 0≤k≤n, the q-binomial coefficient is defined by
(3)[nk]q≔[n]q![k]q![n-k]q!.
In addition, we employ the notation:
(4)(a-x)qn≔∏j=0n-1‍(a-qjx)(n∈ℤ+),(a-x)q∞≔∏j=0∞‍(a-qjx).
For the sequel, it is also convenient to denote
(5)ψq(x)=(1-x)q∞.
In the case 0<q<1, the function ψq is an entire function involved in Euler’s identities (see [13, formulae (9.7) and (9.10)]):
(6)ψq(-x)=∑k=0∞‍qk(k-1)/2xk(1-q)⋯(1-qk),1ψq(x)=∑k=0∞‍xk(1-q)⋯(1-qk)for|x|<1.
For 0<q<1,q-analogues of the exponential function are given by
(7)eq(x)≔∑k=0∞‍xk[k]q!,|x|<11-q,Eq(x)=∑k=0∞‍qk(k-1)/2xk[k]q!.By the virtue of Euler’s identities,(8)eq(x)=∏j=0∞‍(1-(1-q)xqj)-1,|x|<11-q,Eq(x)=∏j=0∞‍(1+(1-q)xqj),
whence
(9)eq(x)Eq(-x)=1.
Clearly, for q=1, we have
(10)[k]1=k,[k]1!=k!,e1(x)=E1(x)=ex.Definition 1.
Givenq∈(0,1), the limit q-Bernstein operator on C[0,1] is defined by f↦Bqf, where
(11)(Bqf)(x)=Bq(f;x)≔{Eq(-x1-q)·∑k=0∞‍f(1-qk)xk(1-q)k[k]q!ifx∈[0,1),f(1)ifx=1,={(1-x)q∞·∑k=0∞‍f(1-qk)(1-q)⋯(1-qk)xkifx∈[0,1),f(1)ifx=1.
Since
(12)(1-x)q∞∑k=0∞‍xk(1-q)⋯(1-qk)=1for|x|<1,
it follows that Bq is a bounded positive linear operator on C[0,1] with ∥Bq∥=1. It can be readily seen from the definition that Bq possesses the end-point interpolation property:
(13)Bq(f;0)=f(0),Bqf(1)=f(1).
It is commonly known in the field that Bq leaves invariant linear functions and maps a polynomial of degree m to a polynomial of degree m (see also Theorem 26). Additional properties of this operator will be considered in the present paper. Prior to presenting the results on Bq, it is worth discussing the origin of the operator itself.


## 2.q-Bernstein Polynomials

This section describes the relation between the limitq-Bernstein operator and the theory of q-Bernstein polynomials. Within the framework of this theory, Bq emerges as a limit for a sequence of the q-Bernstein polynomials. These polynomials were introduced by Phillips in 1997 (cf. [14]) who initiated researches in the area. The summary of the results obtained by Phillips and his collaborators is presented in [15, Ch. 7].Definition 2 (see [14]).
Theq-Bernstein polynomial of f is
(14)Bn,q(f;x)=∑k=0nf([k]q[n]q)pnk(q;x),n=1,2,…,
where
(15)pnk(q;x)≔[nk]qxk∏j=0n-1-k(1-qjx),k=0,1,…n.
Note thatBn,1(f;x) are classical Bernstein polynomials.
Some of the properties of the classical Bernstein polynomials are known to have been taken after by theq-Bernstein polynomials (see [15]). For example, the q-Bernstein polynomials possess the end-point interpolation property, leave invariant linear functions, admit representation with the help of q-differences, and are degree-reducing on polynomials. Apart from that, the q-Bernstein basic polynomials (15) admit a probabilistic interpretation via q-binomial distribution (see [1, 16, 17]). A comprehensive review of the results on the q-Bernstein polynomials along with an extensive bibliography and a collection of open problems on the subject have all been provided in [18]. Recently, modifications of the q-Bernstein polynomials related to the q-Stirling numbers, q-integral representations, and the p-adic numbers have been investigated by Kim et al. in [19–22].
However, further investigation of theq-Bernstein polynomials demonstrates that their convergence properties are essentially different from those of the classical ones and that the cases 0<q<1 and q>1 are different from one another—a difference whose origin can be traced back to the fact that while, for 0<q<1, the q-Bernstein polynomials are positive linear operators on C[0,1], this is no longer valid for q>1.
The next theorem shows the limitq-Bernstein operator rising naturally when a sequence of the q-Bernstein polynomials in the case as 0<q<1 is considered.Theorem 3 (see [23]).
Letq∈(0,1).(i)
Then, for anyf∈C[0,1],
(16)Bn,q(f;x)⟶Bq(f;x)asn⟶∞,
uniformly for x∈[0,1].(ii)
The equalityBq(f;x)=f(x) for x∈[0,1] holds if and only if f is a linear function.Remark 4.
Wang observed [24] that if {Mn,q(f;x)},q∈(0,1) is a sequence of the q-Meier-König and Zeller operator considered by Trif (cf. [25]), then for any f∈C[0,1],
(17)Mn,q(f;x)⟶Bq(f;x)asn⟶∞,
uniformly for x∈[0,1].
It should be emphasized that various analogues of Theorem3 have been proved for different classes of q-operators, as, for example, in [6, 7, 9, 10]. On the top of that, this theorem has triggered the start of further research on the Korovkin-type theorems (cf. [12, 26]). As it turns out, while many q-versions of the known operators—in particular, q-Bernstein polynomials—do not satisfy the conditions of the Korovkin theorem, they do satisfy the conditions of Wang’s Korovkin-type theorem (Theorem 5), which guarantees their uniform convergence on [0,1] to the limit operator.Theorem 5 (see [12]).
LetLn be a sequence of positive linear operators on C[0,1] satisfying the following conditions:(a)
the sequence{Ln(t2;x)} converges uniformly on [0,1],(b)
the sequence{Ln(f;x)} is nondecreasing in n for any convex function f and any x∈[0,1].
Then, there exists an operatorL on C[0,1] such that
(18)Ln(f;x)⟶L(f;x)on[0,1]asn⟶∞,
uniformly on [0,1].Remark 6.
In general, condition (b) cannot be left out completely. The corresponding example is provided in [12, Theorem 1].
Meanwhile, statement (ii) of Theorem3 is a general property of positive linear operator as stated by the next theorem.Theorem 7 (see [10]).
LetL be a positive linear operator on C[0,1] which reproduces linear functions. If L(t2;x)>x2 for x∈(0,1), then Lf=f if and only if f is linear.


## 3. Probabilistic Approach

Another approach toBq is given in terms of probability theory.Consider a functionφ(x) with the positive Taylor coefficients analytic in the disc {x:|x|<r},0<r≤∞,
(19)φ(x)=∑k=0∞‍akxk,a0=1,ak>0,
and consider a random variable ξx(0≤x≤r), whose values do not depend on x and are taken with the following probabilities:
(20)P{ξx=αk}=akxkφ(x)=:pk(x),k=0,1,….LetX be the linear space of functions defined on {αk} so that for f∈X,x∈[0,r), the mathematical expectation E[f(ξx)] exists. We define a linear operator Aφ on X as follows:
(21)(Aφf)(x)≔E[f(ξx)]=∑k=0∞f(αk)pk(x).Suppose that the probability distribution ofξx satisfies the following conditions:(i)
E[ξx]=x, that is, Aφ leaves invariant linear functions,(ii)
E[ξx2]=qx2+bx+c, that is, Aφ takes a square polynomial to a square polynomial.Example 8.
The Poisson distribution with parameterx.Theorem 9 (see [2]).
Letξx be a random variable whose distribution
(22)P{ξx=αk}=akxkφ(x),k=0,1,2,…
satisfies the conditions above. Then,
(23)c=0,q>-1,αk=b1-qk1-q,c=0,ak=(1-q)kbk(1-q)⋯(1-qk),
and the function φ has the form:
(24)φ(x)=∑k=0∞(1-q)kxkbk(1-q)⋯(1-qk).The Theorem means that conditions (i) and (ii) imply a rather specific form of probability distribution.Consider the following particular cases.(1)
Letq=b=1. Then,
(25)φ(x)=∑k=0∞xkk!=ex,αk=k,P{ξx=k}=xkk!e-x,
therefore, ξx has the Poisson distribution with parameter x. Correspondingly,
(26)(Aφf)(x)=∑k=0∞f(k)xkk!e-x.(2)
Forq=1,b=1/n, we obtain
(27)φ(x)=∑k=0∞‍(nx)kk!=enx,αk=kn,P{ξx=kn}=(nx)kk!e-nx.In this case,
(28)(Aφf)(x)=∑k=0∞‍f(kn)(nx)kk!e-nx=Sn(f;x),
that is, Aφ coincides with the Szász-Mirakyan operator. By Feller’s Lemma [27, v. II, Ch. VII, Section 1, Lemma 1], if f∈C[0,∞) is bounded, then Sn(f;x)→f(x) as n→∞, uniformly on any compact subset of [0,∞).(3)
Let0<q<1,b=1-q. Then,
(29)φ(x)=∑k=0∞xk(1-q)⋯(1-qk)=1ψq(x),|x|<1.Besides, αk=1-qk and
(30)P{ξx=1-qk}=ψq(x)xk(1-q)⋯(1-qk).
Therefore,
(31)(Aφf)(x)=ψq(x)∑k=0∞f(1-qk)(1-q)⋯(1-qk)xk=Bq(f;x).
As we can see, in this way, Bq occurs as an analogue of the Szász-Mirakyan operator.


## 4. Approximation Properties ofBq

The approximation by operatorBq was first studied by Videnskii in [28]. Let us recollect that the modulus of continuity of a function f on [0,1] is defined by
(32)ω(f;t)≔sup{|f(x)-f(y)|:|x-y|≤t,x,y∈[0,1]}.The following estimates are valid.Theorem 10 (see [28]).
(i) If f∈C[0,1], then
(33)|Bq(f;x)-f(x)|≤2ω(f;121-q).
Consequently,
(34)Bq(f;x)⟶f(x)asq⟶1-,uniformlyforx∈[0,1].(ii) If f∈C(2)[0,1], then
(35)|Bq(f;x)-f(x)-1-q2f′′(x)x(1-x)|≤K(1-q)x(1-x)ω(f′′;1-q),
where K is a positive constant.
Consequently, forf∈C(2)[0,1],
(36)limq→1-Bq(f;x)-f(x)1-q=f′′(x)2x(1-x),
uniformly on [0,1].The elaboration of these results has been carried out in [29]. Videnskii [28] has also considered the modification of the limit q-Bernstein operator defined for f∈C(2)[0,1] by
(37)B~q(f;x)≔Bq(f;x)-1-q2x(1-x)Bq(f′′;x)
and proved that
(38)|B~q(f;x)-f(x)|≤K(1-q)x(1-x)ω(f′′;1-q),K>0.In [30], Mahmudov has introduced a generalization of the limit q-Bernstein operator defined on the space Cr[0,1] of the r times continuously differentiable functions and proved that, for r≥1, these operators provide a better degree of the approximation than operators Bq, corresponding to r=0.The approximation of the analytic functions in complex domains by the limitq-Bernstein operator has been investigated in [31], where the following results have been established.Theorem 11.
Letf∈C[0,1] admit an analytic continuation from [0,1] into {z:|z-1|<1+ε}. Then, for any compact set K⊂D(ε),
(39)Bq(f;z)⟶f(z),q⟶1-,uniformlyonK.Corollary 12.
Iff is an entire function, then, for any compact set K⊂ℂ,
(40)Bq(f;z)⟶f(z),q⟶1-,uniformlyonK.Finally, we provide an estimate for the rate of approximation for functions analytic inD(r),r>1.Theorem 13.
Letf(z) be analytic in a closed disk D(r)¯ with r>1. Then, for z∈D(r)¯, we have
(41)|Bq(f;z)-f(z)|≤Cf,r(1-q).Remark 14.
Clearly, Corollary12 can also be derived from Theorem 13. Moreover, we obtain that the order of approximation for analytic functions equals (1-q). Using the growth estimates for f, we can estimate Cf,r for r>1.


## 5. Functional-Analytic Properties of the Limitq-Bernstein Operator

To begin with, let us identify the kernel and the image of the limitq-Bernstein operator. The relevant results have been supplied in [23, 32].Theorem 15.
(i) ker Bq={f∈C[0,1]:f(1-qk)=0forallk∈ℤ+} and (ii) imBq={f∈C[0,1]:f(x)=∑k=0∞‍akxk,where∑k=0∞‍akconverges.}Corollary 16.
The image of the limitq-Bernstein operator Bq:C[0,1]→C[0,1] is nonclosed.We say that an operatorT:X→Y is bounded below on a subspace L⊂X if there exists a constant c>0 such that ||Tx||≥c||x|| for each x∈L. An easy consequence of Theorem 15 is that Bq is not bounded below on any subspace which does not contain isomorphic copies of c0.However, for subspaces containing subspaces isomorphic toc0, the situation can be different. To be specific, the following result holds.Theorem 17 (see [33]).
There exists a subspace ofC[0,1] isomorphic to c0 such that the restriction of Bq to this subspace is an isomorphic embedding.Further properties of the image of the limitq-Bernstein operator are expressed by the uniqueness theorems below.In general, for a functionf∈C[0,1], its image under Bq depends on q. Plain calculations show that
(42)Bq(t2;x)=x2+(1-q)x(1-x),
which implies that Bq1(t2;x)≢Bq2(t2;x) for distinct q1 and q2. However, if f is a linear function, then Bq(f;x)=f(x) regardless of q. It is not difficult to see that the converse statement is also true.Theorem 18 (see [32]).
If, for anyq1,q2∈(0,1), we have
(43)Bq1(f;x)≡Bq2(f;x),x∈[0,1],
then f is a linear function.A stronger assertion may be proved for the images of analytic functions.Theorem 19.
Letf be analytic on [0,1]. If, for q1≠q2,
(44)Bq1(f;x)≡Bq2(f;x),x∈[0,1],
then f is a linear function.A closer look can show that this result appears to be sharp and that the statement ceases to be true for infinitely differentiable functions.Now, let us draw attention to the behavior of the iterates of the limitq-Bernstein operator, which have been studied in [34]. By L, we denote the operator of linear interpolation at 0 and 1, that is,
(45)L(f;x)≔(1-x)f(0)+xf(1).Theorem 20 (see [34]).
If{jn} is a sequence of positive integers such that jn→∞, then, for any f∈C[0,1],
(46)Bqjn(f;x)⟶L(f;x)forx∈[0,1]asn⟶∞,
uniformly on [0,1].As an immediate consequence of this theorem, we obtain the following statement mentioned in Section2.Corollary 21.
Letq∈(0,1). Then, Bq(f)=fif and only iff=L(f),thatis,f is a linear function.


## 6. The Improvement of Analytic Properties under the Limitq-Bernstein Operator

Generally speaking, it can be stated thatBq improves the analytic properties of functions. The first result in this direction is the following:Theorem 22 (see [23, 35]).
(i) For any f∈C[0,1], the function Bq(f;x) is continuous on [0,1] and admits an analytic continuation into the open unit disc {z:|z|<1}.
(ii) If f is m(m≥0) times differentiable from the left at 1 and f(m) satisfies the Hölder condition at 1, that is,
(47)|f(m)(x)-f(m)(1)|≤M|x-1|α,M>0,α∈(0,1],
then Bq(f;x) admits an analytic continuation into the disc {z:|z|<q-(m+α)}.
In particular, iff is infinitely differentiable from the left at 1, then Bq(f;z) is an entire function.Remark 23.
In general, an analytic continuation ofBq(f;x) may not be continuous in the closed unit disc.
For a functionF, analytic in a disc {z:|z|≤r}, we denote
(48)M(r;F)≔max|z|≤r|F(z)|.Theorem 24 (see [36]).
(i) If f is analytic at 1, then Bq(f;z) is an entire function and
(49)M(r;Bqf)≤Crmψq(-r),forC,m>0,r≥1.
(ii) If f is analytic in {z:|z-1|<2+ε}, then
(50)M(r;Bqf)≤Cψq(-r),forsomeC>0.Note that(51)C1exp{ln2(r/q)2ln(1/q)}≤ψq(-r)≤C2exp{ln2(r/q)2ln(1/q)}.Therefore, for any entire functionf, the growth of Bq(f;z) does not exceed the growth of ψq(z), showing that for an entire function, whose growth is faster than that of ψq(z), the growth of Bqf is slower than that of f. In other terms, the application of Bq to entire functions slows down a rather speedy growth. It turns out that the same phenomenon occurs for all transcendental entire functions regardless of their growth.Theorem 25 (see [36]).
Iff is a transcendental entire function, then
(52)M(r;Bqf)=o(M(r;f))asr⟶∞.Finally, we state the following noteworthy property of theq-Bernstein operator: it maps binomial (1-x)m to the corresponding q-binomial (x;q)m.Theorem 26 (see [36]).
Iff is a polynomial of degree m, then Bq(f;x) is also a polynomial of degree m. In addition, the following identity holds
(53)(Bq)((1-x)m)=(1-x)(1-qx)⋯(1-qm-1x),m=0,1,2,….The results above indicate how the analytic properties off are transformed under Bq. If f at least satisfies the Hölder condition at 1, then, on the whole, it gets “better", unless f is a polynomial, that is, “too good" to be improved.The results above can be concluded in the form of a table as follows:f(m)∈ Lip αat1⇒Bqf admits an analytic continuation into {z:|z|<q-(m+α)},f infinitely differentiable at 1 ⇒Bqf is entire,fanalyticat1⇒Bqf is entire with M(r;Bqf)≤Craexp(Cln2r),f transcendental entire ⇒Bqf is transcendental entire with M(r;Bqf)≤Cr-u(r)exp(Cln2r),u(r)→+∞asr→∞andM(r;Bqf)=o(M(r;f)), r→∞,f polynomial, degf=m⇒Bqf polynomial, degBqf=m.One can establish that, to a certain extent, the analytic properties off may be retrieved from those of Bqf. For details, see [37]. Put differently, all “⇒" can be replaced with “⇔" provided that we consider the following equivalence relation on C[0,1]:
(54)f~g⟺f(1-qk)=g(1-qk),k∈Z+.
Obviously,
(55)f~g⟺Bqf=Bqg.Then, what happens under the application ofBq to continuous functions—those which do not satisfy the Hölder condition on [0,1]? In this case, Bqf is a function in C[0,1] which possesses an analytic continuation into the open unit disc, and, as a result, the possible lack of smoothness on [0,1) will be corrected by Bq. One can also inquire about the smoothness at 1. In response to this query, it has been shown that, under some minor restrictions, the operator Bq speeds up the convergence of f(x) to f(1) as x→1-. The rate of f(x) approaching f(1) is measured by the localmodulus of continuity at 1:
(56)Ω(f;δ)≔max1-δ≤x≤1|f(x)-f(1)|.Theorem 27 (see [36]).
Iff∈C[0,1] and Ω(f;δ) satisfies the following regularity condition:
(57)∃b∈(0,1),limδ→0+δ∫b1/δ1(Ω(f;t)/t)dtΩ(f;δ)=0,
then Ω(Bqf;δ)=o(Ω(f;δ)) as δ→0+.Corollary 28.
IfC1δβ≤Ω(f;δ)≤C2(ln(1/δ))-α,0<β<α<1, then Ω(Bqf;δ)=o(Ω(f;δ)) as δ→0+.Remark 29.
The condition (57) is rather general. For example, it holds for the functions:
(58)Ω(δ)=δα(ln1δ)β1(ln21δ)β2⋯(lnn1δ)βn,0<α<1,β1,…,βn∈ℝ,n∈ℕ,Ω(δ)=(lnk1δ)-α(lnk+11δ)β1⋯(lnk+j1δ)βj,α>0,β1,…,βj∈ℝ,k,j∈ℕ.However, as it is shown in [38], there exist functions without the Hölder conditions at 1 which do not satisfy (57) such that for some c>0,
(59)Ω(Bqf;δ)≥cΩ(f;δ),δ∈[0,1].


## 7. Concluding Remarks

The limitq-Bernstein operator has remained under scrutiny, and new researches on the subject appear on a regular basis. The aim of the present survey has been not only to exhibit the results related to this operator but also to primarily demonstrate the interrelations of the operator with a variety of mathematical disciplines.Finally, it is beneficial to formulate an open problem for future investigation.Problem. (Eigenvalues and eigenfunctions of the limit q-Bernstein operator). Find all f∈C[0,1] so that
(60)Bqf=λf,λ∈ℂ∖{0}.Conjecture. If Bqf=λf,λ≠0, then f is a polynomial and λ∈{qm(m-1)/2}m=0∞.Comment. The conjecture has been proved under some additional conditions on the smoothness of f at 1 (e.g., for f satisfying the Hölder condition of order α) in [36, Corollary 5.6].



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