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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, $p_0<2$ and $p_1>2$, such that this family isa Schauder basis of $L_s(0,1)$ for all $s>1$ and $p\in[p_0,p_1]$.

Generalised cosine functions, basis and regularity properties