On the density of polynomials in some $L^2(M)$ spaces
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 $Π= \{(x,ϕ):\ x\in \mathbb{R}, ϕ\in [-π,π) \} $, and power-trigonometric polynomials: $p(x,ϕ) = \sum_{m=0}^\infty \sum_{n=-\infty}^\infty α_{m,n} x^m e^{inϕ}$, $α_{m,n}\in \mathbb{C}$, where all but finite number of $α_{m,n}$ 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.