To Determinate Moment Problems

It has been proved that algebraic polynomials P are dense in the space Lp(R, dμ), p∈ (0,∞), iff the measure μ is representable as dμ = wp dν with a finite non-negative Borel measure ν and an upper semi-continuous function w : R → R+ : = [0,∞) such that P is a dense subset of the space C0 w : = {f ∈ C(R) : w(x)f(x) → 0 as |x| → ∞} equipped with the seminorm ‖f‖w := supx∈R w(x)|f(x)|. The similar representation (1 + x2)dμ = w2dν ( (1 + x)dμ = w2dν) with the same ν and w (w(x) = 0, x < 0, and P is also a dense subset of C0 √ x ·w ) corresponds to all those measures (supported by R+) that are uniquely determined by their moments on R (R+). The proof is based on de Branges’ theorem (1959) on weighted polynomial approximation. A more general question on polynomial denseness in a separable Fréchet space in the sense of Banach LΦ(R, dμ) has also been examined.