scholarly journals The Functional-Analytic Properties of the Limitq-Bernstein Operator

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Sofiya Ostrovska
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Dimensional Subspace ◽  
Probabilistic Interpretation ◽  
Bernstein Operator ◽  
Approximation Properties ◽  
Shape Preserving ◽  
Infinite Dimensional ◽  
Boson Theory ◽  
Infinite Dimensional Subspace

The limitq-Bernstein operatorBq,0<q<1, emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution. The latter is used in theq-boson theory to describe the energy distribution in aq-analogue of the coherent state. Lately, the limitq-Bernstein operator has been widely under scrutiny, and it has been shown thatBqis a positive shape-preserving linear operator onC[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 ofBqare studied. Our main result states that there exists an infinite-dimensional subspaceMofC[0,1]such that the restrictionBq|Mis an isomorphic embedding. Also we show that each such subspaceMcontains an isomorphic copy of the Banach spacec0.

On the properties of the limit q-Bernstein operator

2011 ◽  
Vol 48 (2) ◽  
pp. 160-179 ◽  
Author(s):  
Sofiya Ostrovska
Keyword(s):  
Entire Function ◽  
Linear Operator ◽  
Energy Distribution ◽  
Slow Growth ◽  
Probabilistic Interpretation ◽  
Bernstein Operator ◽  
Approximation Properties ◽  
Shape Preserving ◽  
Boson Theory ◽  
The Impact

The limit q-Bernstein operator Bq = B∞,q: C[0, 1] → C[0, 1] emerges naturally as a q-version of the Szász-Mirakyan operator related to the q-deformed Poisson distribution. The latter is used in the q-boson theory to describe the energy distribution in a q-analogue of the coherent state.The limit q-Bernstein operator has been widely studied lately. 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, behavior of iterates, and the impact on the smoothness have been examined.In this paper, it is shown that the possibility of an analytic continuation of Bqf into {z: |z| < R}, R > 1, implies the smoothness of f at 1, which is stronger when R is greater. If Bqf can be extended to an entire function, then f is infinitely differentiable at 1, and a sufficiently slow growth of Bqf implies analyticity of f in {z: |z − 1| < δ}, where δ is greater when the growth is slower. Finally, there is a bound for the growth of Bqf which implies f to be an entire function.

scholarly journals A Survey of Results on the Limit -Bernstein Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Sofiya Ostrovska
Keyword(s):  
Approximation Theory ◽  
Probabilistic Interpretation ◽  
Bernstein Operator ◽  
Approximation Properties ◽  
Shape Preserving ◽  
The Past ◽  
Boson Theory ◽  
Complete Bibliography ◽  
The Impact ◽  
Exemplary Model

The limit -Bernstein operator emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution, which is used in the -boson theory to describe the energy distribution in a -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 -operators. Over the past years, the limit -Bernstein operator has been studied widely from different perspectives. It has been shown that is a positive shape-preserving linear operator on with . 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 -Bernstein operator related to the approximation theory. A complete bibliography is supplied.

Necessary and Sufficient Conditions for the Equality of L(f) and l1

1982 ◽  
Vol 34 (2) ◽  
pp. 406-410 ◽  
Author(s):  
Waleed Deeb
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Sufficient Conditions ◽  
Dimensional Subspace ◽  
Locally Convex Space ◽  
Natural Question ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Infinite Dimensional Subspace

Introduction. Let f be a modulus, ei = (δij) and E = {ei, i = 1, 2, …}. The L(f) spaces were created (to the best of our knowledge) by W. Ruckle in [2] in order to construct an example to answer a question of A. Wilansky. It turned out that these spaces are interesting spaces. For example lp, 0 < p ≦ 1 is an L(f) space with f(x) = xp, and every FK space contains an L(f) space [2]. A natural question is: For which f is L(f) a locally convex space? It is known that L(f) ⊆ l1, for all f modulus (see [2]), and l1 is the smallest locally convex FK space in which E is bounded (see [1]). Thus the question becomes: For which f does L(f) equal l1? In this paper we characterize such f. (An FK space need not be locally convex here.) We also characterize those f for which L(f) contains a convex ball. The final result of this paper is to show that if f satisfies f(x · y) ≦ f(x) · f(y) and L(f) ≠ l1 then L(f) contains no infinite dimensional subspace isomorphic to a Banach space.

scholarly journals Sequences and bases in p-Banach spaces

1986 ◽  
Vol 34 (1) ◽  
pp. 87-92
Author(s):  
M. A. Ariño
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dimensional Subspace ◽  
Basic Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Necessary And Sufficient ◽  
Infinite Dimensional Subspace

Necessary and sufficient condition are given for an infinite dimensional subspace of a p-Banach space X with basis to contain a basic sequence which can be extended to a basis of X.

scholarly journals The Fixed Point Property inc0with an Equivalent Norm

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Berta Gamboa de Buen ◽  
Fernando Núñez-Medina
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Dual Space ◽  
Equivalent Norm ◽  
Dimensional Subspace ◽  
Fixed Point Property ◽  
Point Property ◽  
Infinite Dimensional ◽  
Weak Fixed Point Property ◽  
Infinite Dimensional Subspace

We study the fixed point property (FPP) in the Banach spacec0with the equivalent norm‖⋅‖D. The spacec0with this norm has the weak fixed point property. We prove that every infinite-dimensional subspace of(c0,‖⋅‖D)contains a complemented asymptotically isometric copy ofc0, and thus does not have the FPP, but there exist nonempty closed convex and bounded subsets of(c0,‖⋅‖D)which are notω-compact and do not contain asymptotically isometricc0—summing basis sequences. Then we define a family of sequences which are asymptotically isometric to different bases equivalent to the summing basis in the space(c0,‖⋅‖D),and we give some of its properties. We also prove that the dual space of(c0,‖⋅‖D)over the reals is the Bynum spacel1∞and that every infinite-dimensional subspace ofl1∞does not have the fixed point property.

On a family of photon-added deformed coherent states associated with two-parameter deformed boson oscillator algebra

2004 ◽  
Vol 82 (8) ◽  
pp. 623-646 ◽  
Author(s):  
M H Naderi ◽  
M Soltanolkotabi ◽  
R Roknizadeh
Keyword(s):  
Coherent States ◽  
Photon Number ◽  
Dimensional Subspace ◽  
Oscillator Algebra ◽  
Infinite Dimensional ◽  
Quadrature Squeezing ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra ◽  
Two Parameter ◽  
Infinite Dimensional Subspace

By introducing a generalization of the (p, q)-deformed boson oscillator algebra, we establish a two-parameter deformed oscillator algebra in an infinite-dimensional subspace of the Hilbert space of a harmonic oscillator without first finite Fock states. We construct the associated coherent states, which can be interpreted as photon-added deformed states. In addition to the mathematical characteristics, the quantum statistical properties of these states are discussed in detail analytically and numerically in the context of conventional as well as deformed quantum optics. Particularly, we find that for conventional (nondeformed) photons the states may be quadrature squeezed in both cases Q = pq < 1, Q = pq > 1 and their photon number statistics exhibits a transition from sub-Poissonian to super-Poissonian for Q < 1 whereas for Q > 1 they are always sub-Poissonian. On the other hand, for deformed photons, the states are sub-Poissonian for Q > 1 and no quadrature squeezing occurs while for Q < 1 they show super-Poissonian behavior and there is a simultaneous squeezing in both field quadratures.PACS Nos.: 42.50.Ar, 03.65.–w

Montel Subspaces in the Countable Projective Limits of Lp(μ)-Spaces

1989 ◽  
Vol 32 (2) ◽  
pp. 169-176 ◽  
Author(s):  
J. C. Díaz
Keyword(s):  
Closed Subspace ◽  
Dimensional Subspace ◽  
Projective Limit ◽  
Infinite Dimensional ◽  
Complemented Subspace ◽  
Countable Product ◽  
Köthe Space ◽  
Projective Limits ◽  
Infinite Dimensional Subspace

AbstractLet us suppose one of the following conditions: (a) p ≧ 2 and F is a closed subspace of a projective limit (b) p = 1 and F is a complemented subspace of an echelon Köthe space of order 1, Λ(X,β,μ,gk); and (c) 1 > p > 2 and F is a quotient of a countable product of Lp(μn) spaces. Then, F is Montel if and only if no infinite dimensional subspace of F is normable.

An Infinite Dimensional Vector Space of Universal Functions for H∞ of the Ball

2007 ◽  
Vol 50 (2) ◽  
pp. 172-181 ◽  
Author(s):  
Richard Aron ◽  
Pamela Gorkin
Keyword(s):  
Vector Space ◽  
Dimensional Subspace ◽  
Dimensional Vector ◽  
Universal Functions ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Subspace

AbstractWe show that there exists a closed infinite dimensional subspace of H∞(Bn) such that every function of norm one is universal for some sequence of automorphisms of Bn.

Sign in / Sign up

Export Citation Format

Share Document