On the denseness of Jacobi polynomials (Q1774765)

Summary: Let \(\mathbf{X}\) represent either a space \(\mathbf{C} [-1, 1]\) or \(\mathbf{L}_{\alpha, \beta}^p(w)\), \(1 \leq p < \infty\), of functions on \([-1, 1]\). It is well known that \(\mathbf{X}\) are Banach spaces under the sup and the \(p\)-norms, respectively. We prove that there exist the best possible normalized Banach subspaces \(\mathbf{X}_{\alpha, \beta}^k\) of \(\mathbf{X}\) such that the system of Jacobi polynomials is densely spread on these, or, in other words, each \(f \in \mathbf{X}_{\alpha, \beta}^k\) can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Explicit representation for \(f \in \mathbf{X}_{\alpha, \beta}^k\) has been given.