On the denseness of Jacobi polynomials

Source abstract

Let X represent either a space C [−1, 1] or , 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 of X such that the system of Jacobi polynomials is densely spread on these, or, in other words, each can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Explicit representation for has been given.