Codimension of polynomial subspace in L_2 (R,dμ) for discrete indeterminate measure μ (original) (raw)
Let μ be a positive Borel measure on R such that the polynomials belong to L_p (R,dμ). A necessary and sufficient condition is given for the polynomials to be dense in L_p (R,dμ). It reads as follows: the measure μ should be representable as dμ=w^p dv, with a finite Borel measure v and an upper semicontinuous function w:R→[0,1] such that the polynomials belong to and are dense in the space C_0^w={f∈C(R): f(x)w(x)→0 as |x|→∞}. The proof uses the Mergelyan's criterion for the density of polynomials in L_p spaces.
POLYNOMIAL OPTIMIZATION AND THE MOMENT PROBLEM
2012
There are a wide variety of mathematical problems in different areas which are classified under the title of Moment Problem. We are interested in the moment problem with polynomial data and its relation to real algebra and real algebraic geometry. In this direction, we consider two different variants of moment problem.
A study of the Hamburger moment problem on the real line
2017
This thesis contains an exposition of the Hamburger moment problem. The Hamburger moment problem is an interesting question in analysis that deals with finding the existence of a Borel measure representing a given positive semi-definite linear functional. We begin our exposition by constructing orthogonal polynomials associated with a positive definite sequence. Then we discuss the interlacing property of the zeros of these orthogonal polynomials. We proceed by finding a solution to the truncated Hamburger moment problem and then extend the found solution to the complete Hamburger moment problem. After obtaining a solution to the Hamburger moment problem, we address the problem of determinacy of the moment problem. Finally, we discuss a result that proves the density of polynomials with complex coefficients under the assumption that the Carleman’s condition is satisfied.
Density of polynomials in some L2 spaces on radial rays in the complex plane
Linear Algebra and its Applications, 2011
In this manuscript we study necessary and sufficient conditions for the density of the linear space of matrix polynomials in a linear space of square integrable functions with respect to a matrix of measures supported on a set of radial rays of the complex plane. The connection with a completely indeterminate Hamburger matrix moment problem is stated. Vector valued functions associated in a natural way with a function defined in the union of the radial rays are used. Thus, our first aim is the construction of a linear space of square integrable functions with respect to a matrix of measures supported on a set of radial rays and a positive semi-definite matrix acting on the discrete part of the corresponding inner product. An isometric transformation which allows to reduce the problem of density to the case of the real line is introduced. Finally, some examples of such spaces are shown and its completeness is studied in detail.
Multivariate moment problems: geometry and indeterminateness
Annali Della Scuola Normale Superiore Di Pisa Classe Di Scienze, 2006
The most accurate determinateness criteria for the multivariate moment problem require the density of polynomials in a weighted Lebesgue space of a generic representing measure. We propose a relaxation of such a criterion to the approximation of a single function, and based on this condition we analyze the impact of the geometry of the support on the uniqueness of the representing measure. In particular we show that a multivariate moment sequence is determinate if its support has dimension one and is virtually compact; a generalization to higher dimensions is also given. Among the one-dimensional sets which are not virtually compact, we show that at least a large subclass supports indeterminate moment sequences. Moreover, we prove that the determinateness of a moment sequence is implied by the same condition (in general easier to verify) of the push-forward sequence via finite morphisms.
Moment problem in infinitely many variables
Israel Journal of Mathematics, 2016
The multivariate moment problem is investigated in the general context of the polynomial algebra R[x i | i ∈ Ω] in an arbitrary number of variables x i , i ∈ Ω. The results obtained are sharpest when the index set Ω is countable. Extensions of Haviland's theorem [13] and Nussbaum's theorem [28] are proved. Lasserre's description of the support of the measure in terms of the non-negativity of the linear functional on a quadratic module of R[x i | i ∈ Ω] in [21] is shown to remain valid in this more general situation. The main tool used in the paper is an extension of the localization method developed by the third author in [24], [26] and [27]. Various results proved in [24], [26] and [27]
Extremal polynomials with varying measures
International Mathematical Forum, 2007
We investigate the strong asymptotics for L p-extremal polynomials with respect to varying measures on a rectifiable Jordan curve perturbed by a finite Blaschke sequence of point masses outside the curve