A continuous function space with a Faber basis

Let S ⊂ R be compact with #S = ∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (Pn) ∞ n=0 with deg Pn = n such that every f ∈ C(S) has a unique representation f = ∑∞ i=0 αiPi and call such a basis Faber basis. In the special case of S = Sq = {qk; k ∈ N0} ∪ {0}, 0 < q < 1, we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, this polynomials state an example with A(Sq) 6= U(Sq) = C(Sq), where A(Sq) is the so-called Wiener algebra and U(Sq) is the set of all f ∈ C(Sq) which are uniquely represented by its Fourier series.