Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 280314, 8 pages

doi:10.1155/2012/280314

Research Article

The Functional-Analytic Properties of the Limit -Bernstein Operator

Sofiya Ostrovska

Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

Received 31 August 2012; Accepted 8 October 2012

Academic Editor: Dashan Fan

Copyright © 2012 Sofiya Ostrovska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The limit -Bernstein operator , , emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution. The latter is used in the -boson theory to describe the energy distribution in a -analogue of the coherent state. Lately, the limit -Bernstein operator has been widely under scrutiny, and it has been shown that is a positive shape-preserving linear operator on with . 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 are studied. Our main result states that there exists an infinite-dimensional subspace of such that the restriction is an isomorphic embedding. Also we show that each such subspace contains an isomorphic copy of the Banach space .

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 “-deformed Poisson distribution” (see [1, 2]). The latter is used in the -boson theory, which is a -deformation of the quantum harmonic oscillator formalism [3]. Namely, the -deformed Poisson distribution describes the energy distribution in a -analogue of the coherent state [3, 4]. The -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 , 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 -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 . For any , the -integer is defined by

and the -factorial by

Besides, denotes the -analogue of , that is,

while

For , the -analogues of the exponential function are given by (see [5], formulae (9.7) and (9.10))

By Euler’s Identities (cf., e.g., [5], formulae (9.3) and (9.4)),

whence

Clearly, for , we have

Now, for , let be a random variable possessing a discrete distribution with the probability mass function:

When , we recover the classical Poisson distribution with parameter . If is a function defined on , then the mathematical expectation of equals

We notice that in the case , , operator is the classical Szász-Mirakyan operator. Taking and , we arrive at the definition of the limit -Bernstein operator, which, therefore, may be regarded as an analogue of the Szász-Mirakyan operator.

Definition 1.1 (see [6]). Given , the limit -Bernstein operator is defined by , where

Remark 1.2. It has been proved in [6] that, for any , the function is continuous on and admits an analytic continuation into the open unit disc .

Alternatively, the limit -Bernstein operator emerges as a limit for a sequence of the -Bernstein polynomials in the case (see [6–8]). Recently, Wang has shown in [9] that the same operator is the limit for a sequence of -Meyer-König and Zeller operators. The latter operators have been introduced by Trif in [10].

The limit -Bernstein operator has been studied from different perspectives by Charalambides, Il’inskii, Ostrovska, Videnskii, and Wang. It has been shown in [6, 11] that is a positive shape-preserving linear operator on with , which possesses the end-point interpolation property, leaves invariant linear functions, and maps a polynomial of degree to a polynomial of degree . To be more specific, it takes binomial to the corresponding -binomial—that is,

The approximation with the help of 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 on the smoothness of a function conducted in [14] has revealed the following remarkable phenomenon: in general, the limit -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 -Bernstein operator. Our main result is that there exists an infinite-dimensional subspace of such that the restriction is an isomorphic embedding. Also we show that each such subspace contains an isomorphic copy of the Banach space .

2. Functional-Analytic Properties of

Let us recall that the range of an operator is defined as the set . We say that an operator is bounded below on a subspace if there exists a constant such that for each . We say that is bounded below if it is bounded below on . The space consisting of all convergent to zero sequences with the maximum modulus norm is denoted by . Other relevant terminology can be found in [15, 16] or [17].

Proposition 2.1. (i) The range of the limit -Bernstein operator is nonclosed.

(ii) Let be the subspace of consisting of functions , which are linear on the intervals for . Then the restriction of the limit -Bernstein operator to is injective but is not bounded below. There are subspaces of such that the corresponding restrictions of are compact and nuclear, respectively.

(iii) The restriction of to any subspace of , which does not contain a subspace isomorphic to , 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 . Thus, by the Weierstrass theorem, the range of is dense in . If it had been closed, it would have coincided with the whole space , which contradicts Remark 1.2.

(ii) Injectivity of the restriction of to follows immediately from formula (1.11). The same formula implies that the range of the restriction of to coincides with the range of , 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 factors through . This observation is an immediate consequence of the formula (1.11) and the following two observations: (1) the set of restrictions of functions to the sequence is the space of all convergent sequences; (2) this space with the norm is isomorphic to the space . 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 , we conclude that there are many different subspaces of on which the operator is not bounded below.

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

Theorem 2.2. There exists a subspace of isomorphic to such that the restriction of to this subspace is an isomorphic embedding.

Proof. For each finite subset , we introduce a function satisfying the following conditions:

(i), (ii)

It is clear that we may assume that supports of and are disjoint whenever and are. Therefore, for each disjoint sequence , the space spanned by in is isometric to .

Our purpose is to show that we can select subsets in such a way that the sequence is also equivalent to the unit vector basis of . It is clear that

as . It suffices, therefore, to prove the following estimate:

where is a constant that depends only on . To prove the estimate, we need to show that can be chosen in such a way that the functions are almost-disjointly supported with norms bounded below by a positive constant depending only on . To show this, we observe that since , the following equality holds:

where

for all .

We are going to consider only finite subsets consisting of consecutive integers, that is, subsets of the form , where is the least element of and is the number of elements in . For such , the value is between

To prove inequality (2.2) for suitably chosen finite subsets , we use the following simple assertions.

If we fix and let , the maxima of the functions on approach . For each interval of the form , and each , for sufficiently large and an arbitrary , we have on .

Statement can be verified by straightforward calculations. Indeed,

The value of the function at is

Since the limit of this expression as equals 1, , has been proved.

The statement is obvious.

Now we complete the proof of Theorem 2.2 as follows. Combining the claim about the restriction of to with the statement , we get that there exists depending only on such that for each there is satisfying the condition

We use this statement with and get such that . Since, as it can be readily seen, , there is such that for .

Using we establish the existence of such that, for each finite set of the form , the condition holds for each .

After that we use and pick so that for we have

Since , there exists such that for each .

Proceeding in an obvious way, we construct a sequence and an increasing sequence (, for ) so that

Now, straightforward calculations show that, for each , we have

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

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

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

Corollary 2.4. There exists an operator such that is the identity on . So there exists a “stable with respect to small errors” procedure of reconstruction of a function in from its -image, but there are no such procedures for any subspace of containing no subspaces isomorphic to .

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.

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