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

Abstract

The question of the density of polynomials in some spaces L2(M) is studied. The following two variants of the measure M and the polynomials are considered: (1) an N × N matrix-valued nonnegative Borel measure on ℝ and vector-valued polynomials p(x) = (p0(x), p1(x), …, pN−1(x)), where the pj(x) are complex polynomials and N ∈ ℕ (2) a scalar nonnegative Borel measure on the strip Π = {(x, φ): x ∈ ℝ, ϕ ∈ [−π, π)}, and power-trigonometric polynomials $$p(x,\phi ) = \sum\limits_{m = 0}^\infty {\sum\limits_{n = - \infty }^\infty {\alpha _{m,n} x^m e^{in\phi } } } ,\alpha _{m,n} \in \mathbb{C}$$ , where only finitely many αm,n are nonzero. We show that the polynomials are dense in L2(M) if and only if M is the canonical solution of the corresponding moment problem. It should be stressed that we do not impose any additional constraints on the measure, except the existence of moments. Using the known descriptions of the canonical solutions,, we obtain conditions on the density of polynomials in L2(M). Simultaneously, we establish a model for commuting self-adjoint and unitary operators with spectrum of finite multiplicity.