Astron. Astrophys. 348, 38-42 (1999)

3. Rectangular fields

When the field is rectangular, an orthonormal set of functions can be written easily. Here we consider the special case when is a square of length (in some suitable units); any rectangular field can be handled in a similar manner. In the case considered, an orthonormal set of functions is given by

with . The normalization is defined as

The function is not defined. Note that here we use two indices for the set. Cosines must be used in order to have a complete set (see Eqs. (7), (12), and Appendix A). Our problem is solved in terms of the coefficients :

We now observe that the particular choice of the orthonormal set allows us to use fast Fourier transform (FFT) techniques to evaluate Eqs. (17) and (18). The use of FFT makes the direct method very efficient: in particular the method becomes of order . Moreover, several optimized FFT libraries are available.

The optimal truncation for the series (18) is determined by the adopted grid numbers: for a grid of points, should run from 0 to , and from 0 to (this is standard practice for FFT libraries).

© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999
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