Volume of the Set of Uniformly Bounded Polynomials.

摘要

Let omega be a positive weight function on the interval (in brackets: -1,1). Corresponding orthogonal polynomials are denoted by p(sub n)(omega). Let W(sub n, omega) be a subset of R(sup n). The authors will estimate the volume of W(sub n, omega). The set W(sub n, omega) and the problem about its volume appeared in Kashin's paper (K) in connection with his study of the smallest deviation from zero of polynomials with integer coefficient. Upper and lower bounds of (vol W(sub n, omega))(sup 1/n) coinciding up to a multiplicative constant are obtained in the paper for any weight function w of Szego class. The same lower bound was obtained in (K) for weight functions bounded away from zero. Not that the method is simpler than that of (K).