Non-symmetric fast decreasing polynomials and applications

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.