The Behaviour of Legendre And Ultraspherical Polynomials in Lp-Spaces

Abstract

AbstractWe consider the analogue of the λ(p)problem for subsets of the Legendre polynomials or more general ultraspherical polynomials. We obtain the “best possible” result that if 2 < p < 4 then a random subset of N Legendre polynomials of size N4/p-1 spans an Hilbertian subspace. We also answer a question of König concerning the structure of the space of polynomials of degree n in various weighted Lp-spaces.