2011/02/03 by Sergey M. Zagorodnyuk, Sergey M. Zagorodnyuk, Zagorodnyuk, Sergey M.
Mathematics · #Algebraic and Geometric Analysis #Holomorphic and Operator Theory #Meromorphic and Entire Functions #math.FA #msc:44A60
paper · pdf · doi:10.48550/arxiv.1102.0672
20 pages
arxiv created 2011/02/03 · arxiv updated 2011/02/04
In this paper we study the density of polynomials in some L2(M) spaces. Two choices of the measure M and polynomials are considered: 1) a (N× N) matrix non-negative Borel measure on ℝ and vector-valued polynomials p(x) = (p0(x),p1(x),...,pN-1(x)), pj(x) are complex polynomials, N∈ ℕ; 2) a scalar non-negative Borel measure in a strip Π= \(x,ϕ): x∈ ℝ, ϕ∈ [-π,π) \ , and power-trigonometric polynomials: p(x,ϕ) = ∑m=0^∞ ∑n=-∞^∞ αm,n xm einϕ, αm,n∈ ℂ, where all but finite number of αm,n are zeros. We prove that polynomials are dense in L2(M) if and only if M is a canonical solution of the corresponding moment problem. Using descriptions of canonical solutions, we get conditions for the density of polynomials in L2(M). For this purpose, we derive a model for commuting self-adjoint and unitary operators with a spectrum of a finite multiplicity.