Polynomial bases for continuous function spaces

Let S ⊂ R denote a compact set with infinite cardinality and C(S) the set of real continuous functions on S. We investigate the problem of polynomial and orthogonal polynomial bases of C(S). In case of S = {s0, s1, s2, . . .} ∪ {σ}, where (sk)k=0 is a monotone sequence with σ = limk→∞ sk, we give a sufficient and necessary condition for the existence of a so-called Lagrange basis. Furthermore, we show that little q-Jacobi polynomials which fulfill a certain boundedness property constitute a basis in case of Sq = {1, q, q, . . .} ∪ {0}, 0 < q < 1.