Abstract
In this paper we consider quadratic exponential in two main constructions: as finite Gauss sums and in connection with modified discrete Fourier transforms and also as basis of series in shifted Gaussians. We study spectral properties of modified Fourier discrete transforms, its eigenspaces and eigenvectors. For the case of n = 4k an effect of broken symmetry for discrete Fourier transform is discussed. An application of modified discrete Fourier transform to cryptography is proposed. Also series in quadratic exponentials are considered. An application of series in shifted Gaussians to interpolation problem is analyzed. The correctness of finite system approximation is proved and results of good approximations are illustrated by graphs.
Highlights
In this paper we consider some problems in which instead of classical linear exponentialL1(x, y) = exp(axy) quadratic exponentialQ2(x, y) = exp(ax2 + by2 + cxy) (1)works and gives important consequences and results
Quadratic exponential is applied in such different problems and fields as Fresnel waves, Gabor frames, holography, GAUSSIAN computer package and more
We consider a problem of spectral properties of Discrete Fourier Transform (DFT) matrix for which Gauss sum is a trace value
Summary
In this paper we consider some problems in which instead of classical linear exponential. First idea is to use quadratic exponential instead of linear ones was applied by Fresnel to understand the results of Fraunhofer [1, 17]. He introduced famous Fresnel integrals in connection with this topic. In the first section we consider quadratic exponential in finite sums. This is a famous Gauss sum which is a direct generalization of geometric progression n.
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