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REPRESENTATION OF MEASURES WITH POLYNOMIAL DENSENESS IN Lp (R,dµ), 0 <p<∞, AND ITS APPLICATION TO DETERMINATE MOMENT PROBLEMS

By Andrew G. Bakan and Communicated N. Tomczak-jaegermann

Abstract

Abstract. 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) → 0as|x | →∞}equipped with the seminorm ‖f‖w: = supx∈R w(x)|f(x)|. The similar representation (1 + x2)dµ = w2dν ((1+x)dµ = w2dν) withthesameνand w ( w(x)=0,x<0, and P is also a dense subset of C0 √ ) corresponds to all those measures (supported by x · w 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. 1. Introduction an

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.353.4002
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