Criterion of Polynomial Denseness and General Form of a Linear Continuous Functional on the Space $C_w^0$

Abstract

For an arbitrary function $w:\mathbb{R} \to \left[ {0,1} \right]$, we determine the general form of a linear continuous functional on the space $C_w^0$. The criterion for denseness of polynomials in the space $L_2 \left( {\mathbb{R},d\mu } \right)$ established by Hamburger in 1921 is extended to the spaces $C_w^0$.