Exponential fitting for interpolation of oscillatory functions. A numerical approach

Abstract

We develop an entirely numerical approach for the interpolation of the first derivative of an oscillatory function y(x) with frequency ω. Two versions for the number of data at equidistant points xk, k=1,2,…,K are considered: single y(xk) and pair y(xk),y′′(xk), identified by parameter nd with values 1 and 2, respectively. Experimental results show that the accuracies from the new ω-dependent procedure are better than from the polynomial fitting, with a gain which increases with ω. They also show that an approximate value for ω continues to give advantageous results. A special test is reported with respect to the evolution of the error along the interpolation interval. It is found that when nd=1 the errors have the tendency of increasing in amplitude towards the ends, thus following the Runge effect for the interpolation of y(x) but, quite unexpected, when nd=2 the error amplitude is unchanged over the whole interval.