Generalised cosine functions, basis and regularity properties

We examine regularity and basis properties of the family of rescaled p  -cosine functions. We find sharp estimates for their Fourier coefficients. We then determine two thresholds, p0<2p0<2 and p1>2p1>2, such that this family is a Schauder basis of Ls(0,1)Ls(0,1) for all s>1s>1 and p∈[p0,p1]p∈[p0,p1].