O ct 1 99 8 National Academy of Sciences of Ukraine Institute of Mathematics Preprint 98 . 7 Polynomial Approximation in L p ( R , d μ )

For arbitrary w : R → [0, 1] the general form of the continuous linear functionals on the space C 0 w of all functions f continuous on the real line, lim |x|→+∞ w(x)f (x) = 0 , equipped with seminorm ||f || w := sup x∈R w(x)|f (x)| , is found. The weighted analog 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 are established. New descriptions of the Hamburger and Krein classes of entire functions are obtained. Preprint includes the final representation of all those measures µ for which algebraic polynomials are dense in L p (R, dµ). 1 INTRODUCTION This paper is devoted to the weighted polynomial approximation problem on the real line. Let w(x) be a nonnegative function of real values x , such that for each n = 0, 1, 2,. . ., x n w(x) is bounded. In 1924 S.Bernstein [10] asked for conditions on w such that the algebraic polynomials P are dense in the space C 0 w of all functions f continuous on R , satisfying w(x)f (x) → 0 as |x| → +∞ , where C 0 w is equipped with the seminorm ||f || w := sup x∈R w(x)|f (x)| (for a more explicit survey see [1, 30, 32, 40, 41]). In 1937 S. Isumi and T. Kawata [20] showed that if functions w(x) and − log w(e x) are even and convex on the real line, respectively, then algebraic polynomials P are dense in the space C 0 w if and only if R log w(x) 1 + x 2 dx = −∞ .) proved that a necessary and sufficient condition for the density of P in C