Fast Computation of Trigonometric Sums with Applications to the Frequency Analysis of Astronomical Data

Abstract

We present here a simple and straightforward algorithm to compute trigonometric sums of type:

$${F^k}C\left( {{s_n}} \right) = \sum\limits_{m = 1}^M {{f^k}\left( {{t_m}} \right)\cos \left( {2\pi {t_m}{s_n}} \right)} ,{F^k}S\left( {{s_n}} \right) = \sum\limits_{m = 1}^M {{f^k}\left( {{t_m}} \right)\sin \left( {2\pi {t_m}{s_n}} \right)} $$

, for the irregularly spaced time series f (t _m ), m = 1,..., M and for the full grid s _n , = nδs, n = 0,..., N − 1 of frequencies. The similar approach is known from radio astronomy in the context of synthesis imaging. However, when the so-called “gridding” is method to obtain relatively crude interpolations from unequally spaced two dimensional measurements (see Sramek & Schwab (1988)), here we are seeking for high precision numerical algorithm, applicable even in the context of astrometrical data processing.