Spectral analysis of symmetric operators: Application to the Laplace tidal model (Q1287228)

The main purpose of this paper is to obtain a spectral analysis of a selfadjoint operator by means of a finite set of its values. This method seems reasonable and quite natural, since the other possibility is to use the spectral elements of the operator, which are, in general, inaccessible by computation. In fact, the authors deal only with matrices in finite-dimensional Euclidean spaces (with Hilbertian structure); these matrices may be considered as a finite rank approximation of certain infinite dimensional operators. Each function of a selfadjoint operator depends only on values of this function at the operator spectrum. On the other hand, it is also very well known, that not all possible finite rank approximations of the operator (even selfadjoint) yield reasonable approximation of the whole spectrum. And perhaps here is a delicate point of the problem! The main part of this paper is devoted to the rather simple and well known operations on spectral elements of Hermitean or skew-symmetric matrices. The authors give the formula for the spectral resolution of a Hermitean or a skew-symmetric matrix in the form of a linear combination of orthogonal spectral projections. Further they deduce simple consequences such as formulas for the resolution of a function of the matrix into Fourier series of Chebyshev polynomials, for ``windows'' looking into given parts of the spectrum and so on. They discuss the singular value decomposition of rectangular matrices and some applications of the singular values. For the evaluation of Fourier coefficients the authors suggest to use the fast Fourier transform. A detailed explanation is given in the appendix. As an application the authors present the spectral analysis of the finite element approximation of the Laplace tidal model. Also an example of spectral synthesis is given.