Download Advances in Electronics and Electron Physics, Vol. 75 by Peter W. Hawkes (Ed.) PDF

By Peter W. Hawkes (Ed.)

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A qualitative argument for explaining this fact is the following. If we fix the interval where the data are given, the optimum number of data points is obtained when the data are sampled at the Nyquist rate, as shown by the orthogonality of the functions 4” in this case. If we increase the sampling rate, we add more and more points which are less and less linearly independent from the previous ones and which therefore d o not contain significant new information about the function f . A quantitative analysis of the problem follows from the properties of the matrix W < 1/2, Sn, = 2Wsinc[2W(n - m)], (138) which can be obtained from the Gram matrix (134)(with x, given by Eq.

136)) one finds that uk(J’) = (Ok€k)-’eXp{i[X1 f ( N - l)d/2]}&(N, w ;dy/2n). (143) Slepian proved in (1978) that the DPSWF are simultaneously eigenfunctions of an integral and a differential operator (a property analogous to a basic property of PSWF); they are doubly orthogonal in the sense that they are orthogonal both with respect to the scalar product of L z ( - W, W) and with respect to the scalar product of L 2 ( - $ , $ ) ; moreover Uk(N,W ;x) is an even or odd function of x according to the parity of k, has exactly k zeros in the open interval (- W, W ) ,and exactly N - 1 zeros in the interval (-$,$I.

Another problem is Poisson transform inversion (Saleh, 1978; Bertero and Pike, 1986), a problem related to inversion of photon counting distributions for the purpose of obtaining distributions of classical light intensity fluctuations 1 r+m The solution of this problem is unique in L2(0,+ co),as easily follows from the completeness of Laguerre polynomials. Note that the inversion of the Poisson transform is equivalent to the solution of the Stieltjes moment problem in a suitable weighted space with exponential weight.

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