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

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3. Rectangular fields

When the field [FORMULA] is rectangular, an orthonormal set of functions can be written easily. Here we consider the special case when [FORMULA] is a square of length [FORMULA] (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

[EQUATION]

with [FORMULA]. The normalization [FORMULA] is defined as

[EQUATION]

The function [FORMULA] 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 [FORMULA]:

[EQUATION]

We now observe that the particular choice of the orthonormal set [FORMULA] 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 [FORMULA]. 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 [FORMULA] points, [FORMULA] should run from 0 to [FORMULA], and [FORMULA] from 0 to [FORMULA] (this is standard practice for FFT libraries).

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© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999
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