Polynomial approximation in \(L_p(\mathbb R,d\mu)\) (Q2739816)

For arbitrary \(\omega\colon\mathbb R\to[0,1]\) the general form of the continuous linear functionals on the space \(C_{\omega}^0\) of all continuous functions \(f(x)\) such that \(\lim_{|x|\to+\infty} w(x)f(x)=0\) equipped with the seminorm \(\|f\|_{\omega}\colon=\sup_{x\in\mathbb R}w(x)|f(x)|\) is found. The weighted analogue of the Weierstrass polynomial approximation theorem and a new version of M. G. Krein's theorem about partial fraction decomposition of the reciprocal of an entire function is established. New descriptions of the Hamburger and the Krein classes of entire functions are obtained. A representation of all measures \(\mu\) for which the algebraic polynomials are dense in \(L_p(\mathbb R,d\mu)\) is proposed.