Generalised cosine functions, basis and regularity properties

TL;DR

This paper investigates the regularity and basis properties of rescaled p-cosine functions, providing sharp Fourier coefficient estimates and identifying thresholds for their basis behavior in L_s spaces.

Contribution

It introduces new thresholds p_0 and p_1 for which the p-cosine family forms a Schauder basis in L_s(0,1).

Findings

Sharp Fourier coefficient estimates for p-cosine functions.

Identification of p_0<2 and p_1>2 thresholds for basis properties.

The family forms a Schauder basis for all s>1 and p in [p_0,p_1].

Abstract

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 is a Schauder basis of $L_s(0,1)$ for all $s>1$ and $p\in[p_0,p_1]$.