By Peter W. Hawkes (Ed.)
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Additional resources for Advances in Electronics and Electron Physics, Vol. 75
To our knowledge, however, such a problem has not yet been solved either from the theoretical or from the practical point of view. Some interesting results in this direction have been obtained in the case of the finite Hausdorff moment problem (Talenti, 1987). B. Fourier Transform Inversion with Discrete Data As a first example of an inverse problem with discrete data, we consider the problem of determining a function of bounded support when its Fourier transform is given at a finite number of points.
0165. Slepian, (1978) gives asymptotics of the eigenvalues and eigenfunctions, proving, in particular, that as W -, 0 and N -+ co,so that 7tNW -+ c, we have &, @uk(N, w ;wx) $k(c,x), (144) where &(c, x) is the PSWF of order k and Ak the corresponding eigenvalue. If we recall the behaviour of the eigenvalues of PSWF, this property shows that when the number of sampling points in a fixed interval is large, the number of singular values nearly equal to one is approximately equal to the number of sampling points corresponding to the Nyquist rate.
The resulting linear problem is consistent with the general definition given above. If we remember the definition of a generalized moment problem given in Section II,H, we conclude that a linear inverse problem with discrete data is always a finite section of a generalized moment problem. It is obvious that the information about the physical quantity f that can be extracted from Eq. (1 15) is incomplete. If we denote by X, the linear finitedimensional subspace spanned by the functions 4,,, then the data vector g depends only on the orthogonal projection off onto X,.
Advances in Electronics and Electron Physics, Vol. 75 by Peter W. Hawkes (Ed.)