The Functional-Analytic Properties of the Limit 𝑞-Bernstein Operator

Abstract

The limit 𝑞-Bernstein operator 𝐵_𝑞, 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 𝐵_𝑞 is a positive shape-preserving linear operator on 𝐶[0,1] with ‖𝐵_𝑞‖=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 𝐵_𝑞 are studied. Our main result states that there exists an infinite-dimensional subspace 𝑀 of 𝐶[0,1] such that the restriction 𝐵_𝑞|_𝑀 is an isomorphic embedding. Also we show that each such subspace 𝑀 contains an isomorphic copy of the Banach space 𝑐₀.

1. Introduction

The limit 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 𝑆𝑈_𝑞(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]).

Let 𝑞>0. For any 𝑘∈ℤ₊, the 𝑞-integer [𝑘]_𝑞 is defined by [𝑘]_𝑞∶=1+𝑞+⋯+𝑞^𝑘−1(𝑘∈ℕ),[0]_𝑞∶=0 (1.1) and the 𝑞-factorial [𝑘]_𝑞! by [𝑘]_𝑞!∶=[1]_𝑞[2]_𝑞⋯[𝑘]_𝑞(𝑘∈ℕ),[0]_𝑞!∶=1. (1.2) Besides, (𝑥−𝑎)^𝑘_𝑞 denotes the 𝑞-analogue of (𝑥−𝑎)^𝑘, that is, (𝑥−𝑎)^𝑘_𝑞∶=(𝑥−𝑎)𝑥−𝑞𝑎⋯𝑥−𝑞^𝑘−1𝑎(𝑘∈ℕ),(𝑥−𝑎)⁰_𝑞∶=1, (1.3) while (𝑥−𝑎)^∞_𝑞∶=^∞_𝑗=0𝑥−𝑞^𝑗𝑎. (1.4)

For 0<𝑞<1, the 𝑞-analogues of the exponential function are given by (see [5], formulae (9.7) and (9.10)) 𝑒_𝑞(𝑥)∶=^∞_𝑘=0𝑥^𝑘[𝑘]_𝑞!,|𝑥|<1,𝐸1−𝑞_𝑞(𝑥)=^∞_𝑘=0𝑞^{𝑘(𝑘−1)/2}𝑥^𝑘[𝑘]_𝑞!. (1.5)

By Euler’s Identities (cf., e.g., [5], formulae (9.3) and (9.4)), 𝑒_𝑞(𝑥)=11−1−𝑞𝑥^∞_𝑞,|𝑥|<1,𝐸1−𝑞_𝑞(𝑥)=1+1−𝑞𝑥^∞_𝑞, (1.6) whence 𝑒_𝑞(𝑥)𝐸_𝑞(−𝑥)=1. (1.7) Clearly, for 𝑞=1, we have [𝑘]₁=𝑘,[𝑘]₁!=𝑘!,𝑒₁(𝑥)=𝐸₁(𝑥)=𝑒^𝑥. (1.8)

Now, for 𝑏,𝑞>0, let 𝑋_𝑏,𝑞 be a random variable possessing a discrete distribution with the probability mass function: 𝑝𝑏[𝑘]_𝑞=𝐸_𝑞−𝑥𝑏𝑥^𝑘𝑏^𝑘[𝑘]_𝑞!,𝑘∈ℤ₊. (1.9) 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 𝑋_𝑏 equals 𝐄𝑓𝑋_𝑏,𝑞=𝐸_𝑞−𝑥𝑏^∞_𝑘=0𝑓𝑏[𝑘]_𝑞𝑥^𝑘𝑏^𝑘[𝑘]_𝑞!=∶𝐴_𝑏,𝑞𝑓(𝑥). (1.10) We notice that in the case 𝑞=1, 𝑏=1/𝑛, operator 𝐴_𝑏,𝑞 is the classical Szász-Mirakyan operator. Taking 𝑞∈(0,1) and 𝑏=1−𝑞, we arrive at the definition of the limit 𝑞-Bernstein operator, which, therefore, may be regarded as an analogue of the Szász-Mirakyan operator.

Definition 1.1 (see [6]). Given 𝑞∈(0,1),𝑓∈𝐶[0,1], the limit 𝑞-Bernstein operator is defined by 𝑓↦𝐵_𝑞𝑓, where 𝐵_𝑞𝑓(𝑥)=𝐵_𝑞𝐸𝑓;𝑥∶=^𝑞−𝑥⋅1−𝑞^∞_𝑘∑₌₀𝑓1−𝑞^𝑘𝑥^𝑘1−𝑞^𝑘[𝑘]_𝑞!if[),𝑓(1)𝑥∈0,1if𝑥=1. (1.11)

Remark 1.2. It has been proved in [6] that, for any 𝑓∈𝐶[0,1], the function 𝐵_𝑞(𝑓;𝑥) is continuous on [0,1] and admits an analytic continuation 𝐵_𝑞(𝑓;𝑧) into the open unit disc {𝑧∶|𝑧|<1}.

Alternatively, the limit 𝑞-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 limit 𝑞-Bernstein operator has been studied from different perspectives by Charalambides, Il’inskii, Ostrovska, Videnskii, and Wang. It has been shown in [6, 11] that 𝐵_𝑞 is a positive shape-preserving linear operator on 𝐶[0,1] with ‖𝐵_𝑞‖=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, 𝐵_𝑞(1−𝑥)^𝑚=(1−𝑥)^𝑚_𝑞,𝑚∈ℤ₊. (1.12) The approximation with the help of 𝐵_𝑞 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 𝐵_𝑞 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 limit 𝑞-Bernstein operator. Our main result is that there exists an infinite-dimensional subspace 𝑀 of 𝐶[0,1] such that the restriction 𝐵_𝑞|_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 of 𝐵_𝑞

Let us recall that the range 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 subspace 𝐿⊂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 limit 𝑞-Bernstein operator 𝐵_𝑞∶𝐶[0,1]→𝐶[0,1] is nonclosed.
(ii) Let 𝐿 be the subspace of 𝐶[0,1] consisting of functions 𝑓, which are linear on the intervals [1−𝑞^𝑘−1,1−𝑞^𝑘] for 𝑘∈ℕ. Then the restriction of the limit 𝑞-Bernstein operator 𝐵_𝑞∶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 𝐵_𝑞 are compact and nuclear, respectively.
(iii) The restriction of 𝐵_𝑞 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 𝐵_𝑞. Thus, by the Weierstrass theorem, the range of 𝐵_𝑞 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 of 𝐵_𝑞 to 𝐿 follows immediately from formula (1.11). The same formula implies that the range of the restriction of 𝐵_𝑞 to 𝐿 coincides with the range of 𝐵_𝑞, 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 that 𝐵_𝑞 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−𝑞^𝑘}^∞_𝑘=1 is the space of all convergent sequences; (2) this space with the norm sup_𝑘|𝑓(1−𝑞^𝑘)| 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 Proposition 2.1(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 𝐵_𝑞 is not bounded below.

The main result of the present paper states that for subspaces containing subspaces isomorphic to 𝑐₀ the situation can be different.

Theorem 2.2. There exists a subspace of 𝐶[0,1] isomorphic to 𝑐₀ such that the restriction of 𝐵_𝑞 to this subspace is an isomorphic embedding.

Proof. For each finite subset 𝐼⊂ℕ, we introduce a function ℓ_𝐼∈𝐶[0,1] satisfying the following conditions:(i)‖ℓ_𝐼‖=1, (ii)ℓ_𝐼(𝑥)=^{1if𝑥=1−𝑞𝑘}₀^{,𝑘∈𝐼,}_{if𝑥=1−𝑞𝑘,𝑘∈ℕ⧵𝐼.}It is clear that we may assume that supports of ℓ_𝐼1 and ℓ_𝐼2 are disjoint whenever 𝐼₁ and 𝐼₂ are. Therefore, for each disjoint sequence {𝐼_𝑗}^∞_𝑗=1,𝐼_𝑗⊂ℕ, the space spanned by {ℓ_𝐼𝑗}^∞_𝑗=1 in 𝐶[0,1] is isometric to 𝑐₀.
Our purpose is to show that we can select subsets {𝐼_𝑗}^∞_𝑗=1 in such a way that the sequence {𝐵_𝑞(ℓ_𝐼𝑗)} is also equivalent to the unit vector basis of 𝑐₀. It is clear that ^∞_𝑗=1𝑎_𝑗𝐵_𝑞ℓ_𝐼𝑗≤𝐵_𝑞⋅^∞_𝑗=1𝑎_𝑗ℓ_𝐼𝑗≤sup_𝑗𝑎_𝑗, (2.1) as ‖𝐵_𝑞‖=1. It suffices, therefore, to prove the following estimate: ^∞_𝑗=1𝑎_𝑗𝐵_𝑞ℓ_𝐼𝑗≥𝐶_𝑞sup_𝑗𝑎_𝑗, (2.2) where 𝐶_𝑞>0 is a constant that depends only on 𝑞. To prove the estimate, we need to show that {𝐼_𝑗} can be chosen in such a way that the functions {𝐵_𝑞(ℓ_𝐼𝑗)}^∞_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 𝐸_𝑞(−𝑥/(1−𝑞))=(1−𝑥)^∞_𝑞=(1−𝑥)(1−𝑞𝑥)^∞_𝑞, the following equality holds: 𝐵_𝑞ℓ_𝐼(𝑥)=_𝑘∈𝐼𝑥^𝑘(1−𝑥)⋅1−𝑞𝑥^∞_𝑞1−𝑞^𝑘_𝑞=∶_𝑘∈𝐼𝑥^𝑘(1−𝑥)⋅𝑅_𝑘(𝑥), (2.3) where 0<1−𝑞^∞_𝑞≤1−𝑞^∞_𝑞1−𝑞^𝑘_𝑞≤𝑅_𝑘(𝑥)≤11−𝑞^𝑘_𝑞≤11−𝑞^∞_𝑞<∞ (2.4) for all 𝑘∈ℕ,𝑥∈[0,1].
We are going to consider only finite subsets 𝐼 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 (𝐵_𝑞(ℓ_𝐼))(𝑥),𝑥∈[0,1] is between 1−𝑞^∞_𝑞(1−𝑥)_𝑘∈𝐼𝑥^𝑘=∶𝑚_𝑞𝑥^𝑚1−𝑥^𝑑,11−𝑞^∞_𝑞(1−𝑥)_𝑘∈𝐼𝑥^𝑘=∶𝑀_𝑞𝑥^𝑚1−𝑥^𝑑. (2.5) To prove inequality (2.2) for suitably chosen finite subsets {𝐼_𝑗}^∞_𝑗=1, we use the following simple assertions.(a) If we fix 𝑚 and let 𝑑→∞, the maxima of the functions 𝑥^𝑚(1−𝑥^𝑑) 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 𝑥^𝑚(1−𝑥^𝑑)<𝜀 on (0,𝑎).
Statement (a) can be verified by straightforward calculations. Indeed, 𝑥^𝑘1−𝑥^𝑑′=0(if𝑘≥2)⟺𝑥=0or𝑥=𝑥₀∶=^𝑑𝑘.𝑘+𝑑 (2.6) The value of the function at 𝑥₀ is 𝑘𝑘+𝑑^𝑘/𝑑⋅𝑑.𝑘+𝑑 (2.7) 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 Theorem 2.2 as follows. Combining the claim about the restriction of 𝐵_𝑞(ℓ_𝐼) to [0,1] with the statement (a), we get that there exists 𝑏_𝑞>0 depending only on 𝑞 such that for each 𝑚∈ℕ there is 𝑑 satisfying the condition 𝐵_𝑞ℓ_𝐼>𝑏_𝑞for𝐼={𝑚,𝑚+1,…,𝑚+𝑑−1}. (2.8) We use this statement with 𝑚₁=1 and get 𝐼₁={1,…,𝑑₁} such that ‖𝐵_𝑞(ℓ_𝐼1)‖>𝑏_𝑞. Since, as it can be readily seen, (𝐵_𝑞(ℓ_𝐼1))(1)=0, there is 0<𝑎₁<1 such that |(𝐵_𝑞(ℓ_𝐼1))(𝑥)|<𝑏_𝑞/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 |𝐵_𝑞(ℓ_𝐼2)(𝑥)|<𝑏_𝑞/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 𝐵_𝑞ℓ_𝐼2>𝑏_𝑞. (2.9) Since 𝐵_𝑞(ℓ_𝐼2)(1)=0, there exists 𝑎₂∈(𝑎₁,1) such that |𝐵_𝑞(ℓ_𝐼2)(𝑥)|<𝑏_𝑞/2³ for each 𝑥∈[0,1]⧵(𝑎₁,𝑎₂).
Proceeding in an obvious way, we construct a sequence {𝐼_𝑗}^∞_𝑗=1 and an increasing sequence {𝑎_𝑗}^∞_𝑗=1 (𝑎₀=0, 0<𝑎_𝑗<1 for 𝑗∈ℕ) so that 𝐵_𝑞ℓ_𝐼𝑗(𝑥<𝑏)^𝑞2^𝑗+1for𝑥∈[0,1]⧵𝑎_𝑗−1,𝑎_𝑗,max_𝑥∈[0,1]𝐵_𝑞ℓ_𝐼𝑗(𝑥)>𝑏_𝑞. (2.10) Now, straightforward calculations show that, for each {𝑎_𝑗}^∞_𝑗=1∈𝑐₀, we have ^∞_𝑗=1𝑎_𝑗𝐵_𝑞ℓ_{𝐼𝑗𝐶[0,1]}≥sup_𝑗𝑎_𝑗𝑏_𝑞−𝑏_𝑞⋅12²−𝑏_𝑞⋅12³−⋯=sup_𝑗𝑎_𝑗⋅𝑏_𝑞2. (2.11)

Since the space 𝑐₀ is infinite dimensional and nonreflexive, we get the following corollary.

Corollary 2.3. The operator 𝐵_𝑞 is neither weakly compact nor strictly singular (and thus is noncompact).

Let us denote by 𝑀 the subspace of 𝐶[0,1] constructed in Theorem 2.2. Since it is mapped isomorphically by 𝐵_𝑞, the range 𝐵_𝑞(𝑀)⊂𝐶[0,1] is also isomorphic to 𝑐₀. By the well-known result of Sobczyk [19] (see also [16, p.106], and [20]), 𝐵_𝑞(𝑀) is a complemented subspace of 𝐶[0,1].

Corollary 2.4. There exists an operator 𝑍∶𝐶[0,1]→𝐶[0,1] such that 𝑍𝐵_𝑞 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 𝐵_𝑞-image, but there are no such procedures for any subspace of 𝐶[0,1] containing no subspaces isomorphic to 𝑐₀.

Acknowledgment

The author would like to express her sincere gratitude to P. Danesh from the Academic Writing and Advisory Center of Atilim University for his assistance in the preparation of the paper.