Polynomial Density in Lp(R,dµ) and Representation of All Measures Which Generate a Determinate Hamburger Moment Problem

Abstract

For a positive Borel measure µ on ℝ with all finite moments and unbounded support it has been proved that algebraic polynomials are dense in the space \( {L_p}(\mathbb{R},d\mu ),1 \leqslant p < \infty \)if and only if the measure can be represented in the following form:\( d\mu (x) = w{(x)^p}dv(x)\)where v is some finite positive Borel measure on ℝ and w : ℝ → [0,1]is some upper semicontinuous function for which algebraic polynomials are dense in the seminormed space C w 0. Here is derived a similar representation of all measures which generate a determinate Hamburger moment problem.