On the density of polynomials in some $L^2(M)$ spaces

Abstract

In this paper we study the density of polynomials in some $L^2(M)$ spaces. Two choices of the measure $M$ and polynomials are considered: 1) a $(N\times N)$ matrix non-negative Borel measure on $\mathbb{R}$ and vector-valued polynomials $p(x)=(p_0(x),p_1(x),...,p_{N-1}(x))$, $p_j(x)$ are complex polynomials, $N\in \mathbb{N}$; 2) a scalar non-negative Borel measure in a strip $\Pi =\{(x,\varphi ):x\in \mathbb{R},\varphi \in [-\pi ,\pi )\}$, and power-trigonometric polynomials: $p(x,\varphi )={\sum }_{m=0}^{\infty }{\sum }_{n=-\infty }^{\infty }{\alpha }_{m,n}{x}^m{e}^{in\varphi }$, ${\alpha }_{m,n}\in \mathbb{C}$, where all but finite number of ${\alpha }_{m,n}$ are zeros. We prove that polynomials are dense in $L^2(M)$ 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 $L^2(M)$. For this purpose, we derive a model for commuting self-adjoint and unitary operators with a spectrum of a finite multiplicity.

20 pages

References in corpus (1)

• Devinatz's moment problem: a description of all solutions