Abstract：The procedure that the Gaussian quadrature rule is introduced into Fourier transformation calculation is shown in rigorous mathematical deduction based on the Shift sampling DFT (SFT) theory.The conclusion is that a Fourier integral can be approached with high precision by the weighted sum of a series of SFT with different shift parameters.The weigh is a half of the Gaussian quadrature coefficient,and the sampling parameter is a half of the Gaussian quadrature node plus 0.5.This conclusion provides a rigorous theoretical basis for improving the precision of forward modeling of potential fields (gravity and magnetic) in wave-number domain.Since the sufficient conditions of both the SFT theory and the Gaussian quadrature theory are limited real functions,the Gaussian FFT algorithm based on the conclusion are applicable in forward and inverse Fourier transforms of any limited real functions.
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