
Software: 
To fit a tabulated function with a Fourier approximation Fast Fourier Transform methods are used. This type of method factors the Fourier transform into the product of two transforms, if the number M of data points can be factored. Using complex notation, the coefficients are given by 1
M1
2pi We will assume the interval to be [0,2p] so its length is L=2p. Also, the points are assumed to be evenly spaced and M is assumed to be even. Let's call M=M_{0}. If M_{0 }is factorable into M_{0}=H_{0} H_{1 }we write
2p Some manipulation would bring us to the desired factorization: 1 H_{0}1
2pi
H_{1}1
2pi Of course, if H_{1} can be in turn factored, then the second å can be further factored into two other å's. With multivariable functions the program takes advantage of the fact that the grid is rectangular to shorten the computations. An example of a tabulated function in two variables which
can be approximated by the FOURIER TRANSFORMS program is: The correlation coefficient is computed with
å
[f_{estimated}  f_{mean}]^{2
}SQR (
—————————————————————— ) where the summations are over all independent data points. The standard error of estimate is calculated using
å
[f  f_{estimated} ]^{2
}SQR (
———————————————————— ) where the summations are also over all independent data points. The f_{estimated} is found using the approximation obtained. 

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