The least-squares fit of highly oscillatory functions using Eta-based functions

In this paper we examine the possibility of using the Eta functions as a new base for high quality approximations of oscillatory functions with slowly varying weights. We focus on the least squares and piecewise least squares approximation of such functions and compare the results obtained by using Eta-based sets of functions with those obtained by means of the Legendre polynomials and Fourier series. We find out that the accuracies from these are more or less equivalent for small frequencies but they exhibit different behaviors when the frequency is increased: the accuracy worsens for the Legendre polynomials and Fourier series base but it remains bounded for the new base, in accordance with the known properties of the Eta functions. Such an advantage makes the new base quite attractive for being used in many other mathematical contexts where highly oscillatory functions are involved.