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 $(N\times N)$matrixnon-negativeBorelmeasureon$$\mathbb{R}$$ and vector-valued polynomials $p(x)=(p_0(x),p_1(x),\mathrm{.}\mathrm{.}\mathrm{.},p_{N-1}(x))$, $p_j(x)$ are complex polynomials, $N\in \mathbb{N}$; 2) a scalar non-negative Borel measure in a strip $\mathrm{\Pi }=\{(x,\varphi ):x\in \mathbb{R},\varphi \in [-\pi ,\pi )\}$, and power-trigonometric polynomials: $p(x,\phi) = \sum_{m=0}^\infty \sum_{n=-\infty}^\infty \alpha_{m,n} x^m e^{in\phi}$,$\alpha_{m,n}\in $\mathbb{C}$$,whereallbutfinitenumberof$\alpha_{m,n}$ 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.