The Hamburger moment problem and weighted polynomial approximation on discrete subsets of the real line

2008

Proc. Amer. Math. Soc.

Abstract. It has been proved that algebraic polynomials P are dense in the space L p (R, dµ), p ∈ (0, ∞), iff the measure µ is representable as dµ = w p dν with a finite non-negative Borel measure ν and an upper semi-continuous function w :The similar representation (1 + x 2 )dµ = w 2 dν ( (1 + x)dµ = w 2 dν) with the same ν and w ( w(x) = 0, x < 0, and P is also a dense subset of C 0) corresponds to all those measures (supported by R + ) that are uniquely determined by their moments on R (R + ). The proof is based on de Branges' theorem (1959) on weighted polynomial approximation. A more general question on polynomial denseness in a separable Fréchet space in the sense of Banach L Φ (R, dµ) has also been examined.

Section: Letmentioning

confidence: 99%

Representation of measures with polynomial denseness in $\mathbf {L}_{p} (\mathbb {R}, d\mu )$, $0<p<\infty $, and its application to determinate moment problems

2008

Proc. Amer. Math. Soc.

“…In other words, the incompleteness of algebraic polynomials in the space C w 0 leads to their incompleteness on the restriction of this space to a sufficiently sparse set that is the set of all zeros of a certain entire function of minimal exponential type such that all its simple zeros belong to the set S w . In 1998, Borichev and Sodin [7] established an analogous property for the spaces L d p ( , ) R μ in the case where the measure μ is discrete and its support satisfies the condition…”

Section: Main Resultmentioning

confidence: 89%

On the completeness of algebraic polynomials in the spaces L p (ℝ, dμ)

2009

Ukr Math J

We prove that the theorem on the incompleteness of polynomials in the space C w 0 established by de Branges in 1959 is not true for the space L d p ( , ) R μ if the support of the measure μ is sufficiently dense. Preliminary Information and Main Result1.1. De Branges Theorem. Let M + ( ) R be the cone of finite Borel measures on the real axis R , letand unbounded supportthe set of measures μ ∈ * M ( ) R for which supp μ ⊂ R + : = [ 0, + ∞ ), let C( ) R be the linear space of all functions real-valued and continuous on R, let P be the set of all algebraic polynomials with real coefficients, let P s D [ ] be the set of polynomials from P all roots of which are simple and belong to the set D ⊂ R, let B( ) R be the family of Borel subsets of R, let P be the collection of linear topological spaces of real-valued functions on R for which P is a dense subset, and let W * ( ) R be the set of upper-semicontinuous functions w : R → R + such that x n w

“…In 1959 L. de Branges [13] obtained a solution to this problem. A slightly improved version (see [11] and [31]) of his result is ms follows: let s be the family of entire fimctions B of minimal exponential type having real and simple zeros only and let AB denote the set of these zeros. (5) lim log 1/w(A)…”

Section: The Sets •;(R) and W~(r)mentioning

confidence: 99%

Representation of measures with simultaneous polynomial denseness in Lp(R, dμ), 1≤p<∞

Ruscheweyh

2005

Ark. Mat.