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Positivedefiniteness, integral equations and Fourier transforms
, 2000
"... In this paper we define classes of functions which we call positive definite kernel functions and positive definite kernels. The first class may be thought of as a generalization to two dimensions of the classical positive definite functions of BochnerKhinchin type. We study their properties in dep ..."
Abstract

Cited by 3 (2 self)
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In this paper we define classes of functions which we call positive definite kernel functions and positive definite kernels. The first class may be thought of as a generalization to two dimensions of the classical positive definite functions of BochnerKhinchin type. We study their properties in depth and show how the second class arises by considering the associated integral operators. We give necessary and sufficient conditions for the existence of a bilinear expansion of Mercer type and show the analog of Bochner's theorem in the L setting, namely that a function is a positive definite kernel if and only if its Fourier transform is a positive definite kernel. A simple and elegant sufficient condition for compactness of support of positive definite kernels is given, namely that they are compactly supported along the main diagonal. Several corollaries relating compactness of support of the Fourier transform and analyticity are derived.
Local Stationarity Of L
, 2002
"... This paper shows how the sampling theorem relates with the variations along time of the second order statistics of L processes. As a consequence, and mainly due to the positive semidefiniteness of autocorrelation functions, it is possible to conclude if a nonstationary process is locally stationa ..."
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This paper shows how the sampling theorem relates with the variations along time of the second order statistics of L processes. As a consequence, and mainly due to the positive semidefiniteness of autocorrelation functions, it is possible to conclude if a nonstationary process is locally stationary (i.e., if its second order statistics vary slowly along time) by the direct observation of its 2dimension power spectrum or its Wigner distribution. A simple example illustrates how two different strategies for the estimation of autocorrelation functions from a small number of data can lead to opposite results in terms of local stationarity. 1.