Approximation of Fejér partial sums by interpolating functions

Abstract

The interpolating spline or trigonometric polynomial to a function at equally spaced points approximates the Dirichlet partial sums of its Fourier series with accuracy depending only on the neglected coefficients. We show that the Fejér mean of the Dirichlet sums can be approximated by the arithmetic mean of two Fejér trigonometric interpolants, one at the points with even indexes and one at the points with odd indexes, with an error depending only on the neglected Fourier coefficients and it is positive for positive functions. We also consider the case of Fejér spline interpolants and a constructive relation between Hermite and Fejér interpolants.