# The Functional-Analytic Properties of the Limitq-Bernstein Operator

Authors: Sofiya Ostrovska
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

The limitq-Bernstein operator Bq, 0<q<1, emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution. The latter is used in the q-boson theory to describe the energy distribution in a q-analogue of the coherent state. Lately, the limit q-Bernstein operator has been widely under scrutiny, and 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, eigenstructure, and impact on the smoothness of a function have been examined. In this paper, the functional-analytic properties of Bq are studied. Our main result states that there exists an infinite-dimensional subspace M of C[0,1] such that the restriction Bq|M is an isomorphic embedding. Also we show that each such subspace M contains an isomorphic copy of the Banach space c0.


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## Body

## 1. Introduction

Thelimit q-Bernstein operator comes out naturally as an analogue of the Szász-Mirakyan operator, which is related to the Euler probability distribution—also referred to as the “q-deformed Poisson distribution” (see [1, 2]). The latter is used in the q-boson theory, which is a q-deformation of the quantum harmonic oscillator formalism [3]. Namely, the q-deformed Poisson distribution describes the energy distribution in a q-analogue of the coherent state [3, 4]. The q-analogue of the boson operator calculus has proved to be a powerful tool in theoretical physics by providing explicit expressions for the representations of the quantum group SUq(2), which is by now known to play a profound role in a variety of different problems, such as integrable models in the field theory, exactly solvable lattice models of statistical mechanics, and conformal field theory among others. Therefore, properties of the q-deformed Poisson distribution and its related linear operators are of significant interest for applications.In the sequel, the following notations and definitions are employed (cf., e.g., [5]).Letq>0. For any k∈ℤ+, the q-integer[k]q is defined by
(1.1)[k]q:=1+q+⋯+qk-1(k∈ℕ),[0]q:=0
and the q-factorial[k]q! by
(1.2)[k]q!:=[1]q[2]q⋯[k]q(k∈ℕ),[0]q!:=1.
Besides, (x-a)qk denotes the q-analogue of (x-a)k, that is,
(1.3)(x-a)qk:=(x-a)(x-qa)⋯(x-qk-1a)(k∈ℕ),(x-a)q0:=1,
while
(1.4)(x-a)q∞:=∏j=0∞‍(x-qja).For0<q<1, the q-analogues of the exponential function are given by (see [5], formulae (9.7) and (9.10))
(1.5)eq(x):=∑k=0∞‍xk[k]q!,|x|<11-q,Eq(x)=∑k=0∞‍qk(k-1)/2xk[k]q!.By Euler’s Identities (cf., e.g., [5], formulae (9.3) and (9.4)),
(1.6)eq(x)=1(1-(1-q)x)q∞,|x|<11-q,Eq(x)=(1+(1-q)x)q∞,
whence
(1.7)eq(x)Eq(-x)=1.
Clearly, for q=1, we have
(1.8)[k]1=k,[k]1!=k!,e1(x)=E1(x)=ex.Now, forb,q>0, let Xb,q be a random variable possessing a discrete distribution with the probability mass function:
(1.9)p(b[k]q)=Eq(-xb)xkbk[k]q!,k∈ℤ+.
When b=q=1, we recover the classical Poisson distribution with parameter x. If f is a function defined on {b[k]q}k=0∞, then the mathematical expectation of Xb equals
(1.10)E[f(Xb,q)]=Eq(-xb)∑k=0∞‍f(b[k]q)xkbk[k]q!=:(Ab,qf)(x).
We notice that in the case q=1, b=1/n, operator Ab,q is the classical Szász-Mirakyan operator. Taking q∈(0,1) and b=1-q, we arrive at the definition of the limitq-Bernstein operator, which, therefore, may be regarded as an analogue of the Szász-Mirakyan operator.Definition 1.1 (see [6]).
Givenq∈(0,1),f∈C[0,1], the limitq-Bernstein operator is defined by f↦Bqf, where
(1.11)(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.Remark 1.2.
It has been proved in [6] that, for any f∈C[0,1], the function Bq(f;x) is continuous on [0,1] and admits an analytic continuation Bq(f;z) into the open unit disc {z:|z|<1}.Alternatively, the limitq-Bernstein operator emerges as a limit for a sequence of the q-Bernstein polynomials in the case 0<q<1 (see [6–8]). Recently, Wang has shown in [9] that the same operator is the limit for a sequence of q-Meyer-König and Zeller operators. The latter operators have been introduced by Trif in [10].The limitq-Bernstein operator has been studied from different perspectives by Charalambides, Il’inskii, Ostrovska, Videnskii, and Wang. It has been shown in [6, 11] that Bq is a positive shape-preserving linear operator on C[0,1] with ∥Bq∥=1, which possesses the end-point interpolation property, leaves invariant linear functions, and maps a polynomial of degree m to a polynomial of degree m. To be more specific, it takes binomial (1-x)m to the corresponding q-binomial—that is,
(1.12)Bq(1-x)m=(1-x)qm,m∈ℤ+.
The approximation with the help of Bq has been studied in [12], while the properties of its range have been presented in [13]. The probabilistic approaches have been developed in [1, 2]. The investigation of the impact of Bq on the smoothness of a function conducted in [14] has revealed the following remarkable phenomenon: in general, the limit q-Bernstein operator improves the analytic properties of a function, provided the function is neither “very good” (a polynomial) nor “very bad” (without a certain regularity condition).In this paper, we study the functional-analytic properties of the limitq-Bernstein operator. Our main result is that there exists an infinite-dimensional subspace M of C[0,1] such that the restriction Bq|M is an isomorphic embedding. Also we show that each such subspace M contains an isomorphic copy of the Banach space c0.


## 2. Functional-Analytic Properties ofBq

Let us recall that therange of an operator T:X→Y is defined as the set {y∈Y:∃x∈X,Tx=y}. We say that an operator T:X→Y is bounded below on a subspaceL⊂X if there exists a constant c>0 such that ||Tx||≥c||x|| for each x∈L. We say that T:X→Y is bounded below if it is bounded below on X. The space consisting of all convergent to zero sequences with the maximum modulus norm is denoted by c0. Other relevant terminology can be found in [15, 16] or [17].Proposition 2.1.
(i) The range of the limitq-Bernstein operator Bq:C[0,1]→C[0,1] is nonclosed.
(ii) LetL be the subspace of C[0,1] consisting of functions f, which are linear on the intervals [1-qk-1,1-qk] for k∈ℕ. Then the restriction of the limit q-Bernstein operator Bq:C[0,1]→C[0,1] to L is injective but is not bounded below. There are subspaces of L such that the corresponding restrictions of Bq are compact and nuclear, respectively.
(iii) The restriction ofBq to any subspace of C[0,1], which does not contain a subspace isomorphic to c0, is strictly singular and thus is not bounded below.Proof.
(i) It can be readily seen from (1.12) that all polynomials are in the range of Bq. Thus, by the Weierstrass theorem, the range of Bq is dense in C[0,1]. If it had been closed, it would have coincided with the whole space C[0,1], which contradicts Remark 1.2.
(ii) Injectivity of the restriction ofBq to L follows immediately from formula (1.11). The same formula implies that the range of the restriction of Bq to L coincides with the range of Bq, whence it is nonclosed. On the other hand, it is known (see [16, Prop. 2.c.4]) that the condition that the range is nonclosed implies that the operator is not bounded below and that there are subspaces, restrictions to which are compact (even nuclear) operators.
(iii) To begin with, we observe thatBq factors through c0. This observation is an immediate consequence of the formula (1.11) and the following two observations: (1) the set of restrictions of functions f∈C[0,1] to the sequence {1-qk}k=1∞ is the space of all convergent sequences; (2) this space with the norm supk|f(1-qk)| is isomorphic to the space c0. Applying the well-known results on Banach space geometry (see [15, Chapter 2], [16, Chapter 2], [17]), we derive the statement.Combining Proposition2.1(iii) with the classical Banach-Mazur theorem [18, Ch. XI, §8] on the universality of C[0,1], we conclude that there are many different subspaces of C[0,1] on which the operator Bq is not bounded below.The main result of the present paper states that for subspaces containing subspaces isomorphic toc0 the situation can be different.Theorem 2.2.
There exists a subspace ofC[0,1] isomorphic to c0 such that the restriction of Bq to this subspace is an isomorphic embedding.Proof.
For each finite subsetI⊂ℕ, we introduce a function ℓI∈C[0,1] satisfying the following conditions:(i)
∥ℓI∥=1,(ii)
ℓI(x)={1ifx=1-qk,k∈I,0ifx=1-qk,k∈ℕ∖I.It is clear that we may assume that supports of ℓI1 and ℓI2 are disjoint whenever I1 and I2 are. Therefore, for each disjoint sequence {Ij}j=1∞,Ij⊂ℕ, the space spanned by {ℓIj}j=1∞ in C[0,1] is isometric to c0.
Our purpose is to show that we can select subsets{Ij}j=1∞ in such a way that the sequence {Bq(ℓIj)} is also equivalent to the unit vector basis of c0. It is clear that
(2.1)∥∑j=1∞‍ajBq(ℓIj)∥≤∥Bq∥·∥∑j=1∞‍ajℓIj∥≤supj|aj|,
as ∥Bq∥=1. It suffices, therefore, to prove the following estimate:
(2.2)∥∑j=1∞‍ajBq(ℓIj)∥≥Cqsupj|aj|,
where Cq>0 is a constant that depends only on q. To prove the estimate, we need to show that {Ij} can be chosen in such a way that the functions {Bq(ℓIj)}j=1∞ are almost-disjointly supported with norms bounded below by a positive constant depending only on q. To show this, we observe that since Eq(-x/(1-q))=(1-x)q∞=(1-x)(1-qx)q∞, the following equality holds:
(2.3)(BqℓI)(x)=∑k∈I‍xk(1-x)·(1-qx)q∞(1-q)qk=:∑k∈I‍xk(1-x)·Rk(x),
where
(2.4)0<(1-q)q∞≤(1-q)q∞(1-q)qk≤Rk(x)≤1(1-q)qk≤1(1-q)q∞<∞
for all k∈ℕ,x∈[0,1].
We are going to consider only finite subsetsI consisting of consecutive integers, that is, subsets of the form I={m,m+1,…,m+d-1}, where m is the least element of I and d is the number of elements in I. For such I, the value (Bq(ℓI))(x),x∈[0,1] is between
(2.5)(1-q)q∞(1-x)∑k∈I‍xk=:mqxm(1-xd),1(1-q)q∞(1-x)∑k∈I‍xk=:Mqxm(1-xd).
To prove inequality (2.2) for suitably chosen finite subsets {Ij}j=1∞, we use the following simple assertions.(a)
If we fixm and let d→∞, the maxima of the functions xm(1-xd) on [0,1] approach 1.(b)
For each interval of the form(0,a),a<1, and each ε>0, for sufficiently large m∈ℕ and an arbitrary d∈ℕ, we have xm(1-xd)<ε on (0,a).
Statement(a) can be verified by straightforward calculations. Indeed,
(2.6)(xk(1-xd))′=0(ifk≥2)⟺x=0orx=x0:=kk+dd.
The value of the function at x0 is
(2.7)(kk+d)k/d·(dk+d).
Since the limit of this expression as d→∞ equals 1, (a), has been proved.
The statement(b) is obvious.
Now we complete the proof of Theorem2.2 as follows. Combining the claim about the restriction of Bq(ℓI) to [0,1] with the statement (a), we get that there exists bq>0 depending only on q such that for each m∈ℕ there is d satisfying the condition
(2.8)∥Bq(ℓI)∥>bqforI={m,m+1,…,m+d-1}.
We use this statement with m1=1 and get I1={1,…,d1} such that ∥Bq(ℓI1)∥>bq. Since, as it can be readily seen, (Bq(ℓI1))(1)=0, there is 0<a1<1 such that |(Bq(ℓI1))(x)|<bq/22 for x∈[a1,1].
Using(b) we establish the existence of m2 such that, for each finite set I2 of the form {m2,m2+1,…,M}, the condition |Bq(ℓI2)(x)|<bq/23 holds for each x∈(0,a1).
After that we use(a) and pick d2 so that for I2={m2,m2+1,…,m2+d2-1} we have
(2.9)∥Bq(ℓI2)∥>bq.
Since Bq(ℓI2)(1)=0, there exists a2∈(a1,1) such that |Bq(ℓI2)(x)|<bq/23 for each x∈[0,1]∖(a1,a2).
Proceeding in an obvious way, we construct a sequence{Ij}j=1∞ and an increasing sequence {aj}j=1∞ (a0=0, 0<aj<1 for j∈ℕ) so that
(2.10)|Bq(ℓIj)(x)|<bq2j+1forx∈[0,1]∖(aj-1,aj),maxx∈[0,1]|Bq(ℓIj)(x)|>bq.
Now, straightforward calculations show that, for each {aj}j=1∞∈c0, we have
(2.11)∥∑j=1∞‍ajBq(ℓIj)∥C[0,1]≥supj|aj|(bq-bq·122-bq·123-⋯)=supj|aj|·bq2.Since the spacec0 is infinite dimensional and nonreflexive, we get the following corollary.Corollary 2.3.
The operatorBq is neither weakly compact nor strictly singular (and thus is noncompact).Let us denote byM the subspace of C[0,1] constructed in Theorem 2.2. Since it is mapped isomorphically by Bq, the range Bq(M)⊂C[0,1] is also isomorphic to c0. By the well-known result of Sobczyk [19] (see also [16, p.106], and [20]), Bq(M) is a complemented subspace of C[0,1].Corollary 2.4.
There exists an operatorZ:C[0,1]→C[0,1] such that ZBq is the identity on M. So there exists a “stable with respect to small errors” procedure of reconstruction of a function in M from its Bq-image, but there are no such procedures for any subspace of C[0,1] containing no subspaces isomorphic to c0.



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