A continuous function space with a Faber basis

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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 ( P n ) n =0 ∞ with deg P n =n such that every f ∈ C ( S ) has a unique representation f=∑ i=0 ∞ α i P i and call such a basis Faber basis. In the special case of S=S q ={q k ;k∈ N 0 }∪{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, these polynomials state an example with A ( S q )≠ U ( S q )= C ( S q ), where A ( S q ) is the so-called Wiener algebra and U ( S q ) is the set of all f ∈ C ( S q ) which are uniquely represented by its Fourier series.