On the denseness of Jacobi polynomials

Abstract

Let X represent either a space C[−1,1] or Lα,βp(w), 1≤p<∞, of functions on [−1,1]. It is well known that X are Banach spaces under the sup and the p-norms, respectively. We prove that there exist the best possible normalized Banach subspaces Xα,βk of X such that the system of Jacobi polynomials is densely spread on these, or, in other words, each f∈Xα,βk can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Explicit representation for f∈Xα,βk has been given.