A continuous function space with a Faber basis

Abstract

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 iP 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.