On the Bernstein-Walsh-Siciak theorem

Rafał Pierzchała

The search session has expired. Please query the service again.

Displaying similar documents to “On the Bernstein-Walsh-Siciak theorem”

Quasianalytic functions in the sense of Bernstein

W. Pleśniak

Similarity:

CONTENTS1. Introduction................................................................................................................ 32. The extremal function.............................................................................................. 83. Some lemmas on polynomials............................................................................. 124. Category theorems in topological groups........................................................... 165. Best approximation...

Mixed norm condition numbers for the univariate Bernstein basis

Tom Lyche, Karl Scherer (2006)

Banach Center Publications

Similarity:

We study mixed norm condition numbers for the univariate Bernstein basis for polynomials of degree n, that is, we measure the stability of the coefficients of the basis in the $l_{q}$ -sequence norm whereas the polynomials to be represented are measured in the $L_{p}$ -function norm. The resulting condition numbers differ from earlier results obtained for p = q.

The sharpness of convergence results for $q$ -Bernstein polynomials in the case $q > 1$

Sofiya Ostrovska (2008)

Czechoslovak Mathematical Journal

Similarity:

Due to the fact that in the case $q > 1$ the $q$ -Bernstein polynomials are no longer positive linear operators on $C [0, 1],$ the study of their convergence properties turns out to be essentially more difficult than that for $q < 1 .$ In this paper, new saturation theorems related to the convergence of $q$ -Bernstein polynomials in the case $q > 1$ are proved.

Schroeder-Bernstein Quintuples for Banach Spaces

Elói Medina Galego (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let X and Y be two Banach spaces, each isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain necessary and sufficient conditions on the quintuples (p,q,r,s,t) in ℕ for X to be isomorphic to Y whenever ⎧ $X X^{p} \oplus Y^{q}$ , ⎨ ⎩ $Y^{t} X^{r} \oplus Y^{s}$ . Such quintuples are called Schroeder-Bernstein quintuples for Banach spaces and they yield a unification of the known decomposition...

An Inequality for Trigonometric Polynomials

N. K. Govil, Mohammed A. Qazi, Qazi I. Rahman (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

The main result says in particular that if $t (ζ) : = \sum_{ν = - n} ⁿ c_{ν} e^{i ν ζ}$ is a trigonometric polynomial of degree n having all its zeros in the open upper half-plane such that |t(ξ)| ≥ μ on the real axis and cₙ ≠ 0, then |t’(ξ)| ≥ μn for all real ξ.

Singular localization of $𝔤$ -modules and applications to representation theory

Erik Backelin, Kobi Kremnitzer (2015)

Journal of the European Mathematical Society

Similarity:

We prove a singular version of Beilinson–Bernstein localization for a complex semi-simple Lie algebra following ideas from the positive characteristic case settled by [BMR06]. We apply this theory to translation functors, singular blocks in the Bernstein–Gelfand–Gelfand category O and Whittaker modules.

Basic relations valid for the Bernstein spaces $B ²_{σ}$ and their extensions to larger function spaces via a unified distance concept

P. L. Butzer, R. L. Stens, G. Schmeisser (2014)

Banach Center Publications

Similarity:

Some basic theorems and formulae (equations and inequalities) of several areas of mathematics that hold in Bernstein spaces $B_{σ}^{p}$ are no longer valid in larger spaces. However, when a function f is in some sense close to a Bernstein space, then the corresponding relation holds with a remainder or error term. This paper presents a new, unified approach to these errors in terms of the distance of f from $B_{σ}^{p}$ . The difficult situation of derivative-free error estimates is also covered. ...

Bad properties of the Bernstein numbers

Albrecht Pietsch (2008)

Studia Mathematica

Similarity:

We show that the classes $_{p}^{b e r n} : = T : (b ₙ (T)) \in l_{p}$ associated with the Bernstein numbers bₙ fail to be operator ideals. Moreover, $_{p}^{b e r n} \circ_{q}^{b e r n} ⊈_{r}^{b e r n}$ for 1/r = 1/p + 1/q.

Approximation properties of bivariate complex $q$ -Bernstein polynomials in the case $q > 1$

Nazim I. Mahmudov (2012)

Czechoslovak Mathematical Journal

Similarity:

In the paper, we discuss convergence properties and Voronovskaja type theorem for bivariate $q$ -Bernstein polynomials for a function analytic in the polydisc $D_{R_{1}} \times D_{R_{2}} = {z \in C : | z | < R_{1}} \times {z \in C : | z | < R_{1}}$ for arbitrary fixed $q > 1$ . We give quantitative Voronovskaja type estimates for the bivariate $q$ -Bernstein polynomials for $q > 1$ . In the univariate case the similar results were obtained by S. Ostrovska: $q$ -Bernstein polynomials and their iterates. J. Approximation Theory 123 (2003), 232–255. and S. G. Gal: Approximation by Complex Bernstein...

A note on extensions of Pełczyński's decomposition method in Banach spaces

Elói Medina Galego (2007)

Studia Mathematica

Similarity:

Let X,Y,A and B be Banach spaces such that X is isomorphic to Y ⊕ A and Y is isomorphic to X ⊕ B. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In the present paper, we give a necessary and sufficient condition on sextuples (p,q,r,s,u,v) in ℕ with p + q ≥ 2, r + s ≥ 1 and u, v ∈ ℕ* for X to be isomorphic to Y whenever these spaces satisfy the following decomposition scheme: ⎧ $X^{u} \sim X^{p} \oplus Y^{q}$ , ⎨ ⎩ $Y^{v} \sim A^{r} \oplus B^{s}$ . Namely, Ω =...

Sets with the Bernstein and generalized Markov properties

Mirosław Baran, Agnieszka Kowalska (2014)

Annales Polonici Mathematici

Similarity:

It is known that for $C^{\infty}$ determining sets Markov’s property is equivalent to Bernstein’s property. We are interested in finding a generalization of this fact for sets which are not $C^{\infty}$ determining. In this paper we give examples of sets which are not $C^{\infty}$ determining, but have the Bernstein and generalized Markov properties.

Polynomial inequalities on general subsets of $R^{N}$

Pierre Goetgheluck (1989)

Colloquium Mathematicae

Similarity:

Functions of Baire class one

Denny H. Leung, Wee-Kee Tang (2003)

Fundamenta Mathematicae

Similarity:

Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies $β (f) \leq ω^{ξ ₁} \cdot ω^{ξ ₂}$ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1...

On the closure of Baire classes under transfinite convergences

Tamás Mátrai (2004)

Fundamenta Mathematicae

Similarity:

Let X be a Polish space and Y be a separable metric space. For a fixed ξ < ω₁, consider a family $f_{α} : X \to Y (α < ω ₁)$ of Baire-ξ functions. Answering a question of Tomasz Natkaniec, we show that if for a function f: X → Y, the set $α < ω ₁ : f_{α} (x) \neq f (x)$ is finite for every x ∈ X, then f itself is necessarily Baire-ξ. The proof is based on a characterization of $Σ ⁰_{η}$ sets which can be interesting in its own right.

Extension of functions with small oscillation

Denny H. Leung, Wee-Kee Tang (2006)

Fundamenta Mathematicae

Similarity:

A classical theorem of Kuratowski says that every Baire one function on a $G_{δ}$ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this hierarchy depending on its oscillation index β(f). We prove a refinement of Kuratowski’s theorem: if Y is a subspace of a metric space X and f is a...

Typical multifractal box dimensions of measures

L. Olsen (2011)

Fundamenta Mathematicae

Similarity:

We study the typical behaviour (in the sense of Baire’s category) of the multifractal box dimensions of measures on $ℝ^{d}$ . We prove that in many cases a typical measure μ is as irregular as possible, i.e. the lower multifractal box dimensions of μ attain the smallest possible value and the upper multifractal box dimensions of μ attain the largest possible value.

Descriptive properties of elements of biduals of Banach spaces

Pavel Ludvík, Jiří Spurný (2012)

Studia Mathematica

Similarity:

If E is a Banach space, any element x** in its bidual E** is an affine function on the dual unit ball $B_{E *}$ that might possess a variety of descriptive properties with respect to the weak* topology. We prove several results showing that descriptive properties of x** are quite often determined by the behaviour of x** on the set of extreme points of $B_{E *}$ , generalizing thus results of J. Saint Raymond and F. Jellett. We also prove a result on the relation between Baire classes and intrinsic Baire...

An extremal problem for Bernstein polynomials.

Gavrea, Ioan, Kacsó, Daniela (1998)

General Mathematics

Similarity:

One-sided approximation by trigonometric polynomials in $L_{p}$ -norm, 0

D. P. Dryanov (1989)

Banach Center Publications

Similarity:

On the Degre of L1-aproximation by Modified Bernstein Polynomials

Suresh Prasad Singh, O.P. Varshney, Govind Prasad (1983)

Publications de l'Institut Mathématique

Similarity: