Non-symmetric fast decreasing polynomials and applications

摘要

A polynomial P _n is called fast decreasing if P _n(0)=1, and, on [-1, 1], P _n decreases fast (in terms of n and the distance from 0) as we move away from the origin. This paper considers the version when P _n has to decrease only on some non-symmetric interval [-a, 1] with possibly small a. In this case one gets a faster decrease, and this type of extension is needed in some problems, when symmetric fast decreasing polynomials are not sufficient. We shall apply such non-symmetric fast decreasing polynomials to find local bounds for Christoffel functions and for local zero spacing of orthogonal polynomials with respect to a doubling measure close to a local endpoint.