Finite signal reconstruction from its Fourier's spectrum

Regularization theory for the calculation of the Fourier transformations of continuous functions that vanish at infinity is considered. The procedure is based on use of the Gaussian means. It is shown that the Tikhonov regularization method of the first order gives a stabilizing multiplier in the form of rational functions. Regularization of the continuous and discrete and fast multidimensional Fourier transformations are considered as well. The analytical study and computational treatment of the problem are presented.