From high oscillation to rapid approximation I: modified Fourier expansions

In this paper, we consider a modification of the classical Fourier expansion, whereby in [– 1, 1] the sin πnx functions are replaced by $\sin \pi\left(n-\frac{1}{2} \right )x$, n ≥ 1. This has a number of important advantages in the approximation of analytic, nonperiodic functions. In particular, expansion coefficients decay like $\mathscr{O}$(n⁻²), rather than like $\mathscr{O}$(n⁻¹). We explore theoretical features of these modified Fourier expansions, prove suitable versions of the Fejér and de la Vallée Poussin theorems and expand the coefficients into asymptotic series. This expansion is a key towards the computation of expansion coefficients by asymptotic and Filon-type methods. We explore this issue in some detail and present a number of algorithms which require $\mathscr{O}$(m) operations in the computation of the first m expansion coefficients.