This note is related to the famous question raised by Mark Kac and concerning the determination of the shape of the drum by the eigenvalues of its governing equation. Here, we allow the drum to be composed by several different types of membranes and we consider the problem of hearing the composition of the drum, starting from the eigenvalues of numerical approximations of the related equation. Some key tools, taken from asymptotic linear algebra, are reported and extended, and allow somehow to answer to the question in the positive. (2000). Primary 65F10; Secondary 15A18.
Mathematics Subject Classification
We propose an FFT-based algorithm for computing fundamental solutions of difference operators with constant coefficients. Our main contribution is to handle cases where the symbol has zeros.
Given an approximating class of sequences { { Bn, m }n }m for { An }n, we prove that { { Bn, m + }n }m (X+ being the pseudo-inverse of Moore-Penrose) is an approximating class of sequences for { An + }n, where { An }n is a sparsely vanishing sequence of matrices An of size dn with dk > dq for k > q, k, q ∈ N. As a consequence, we extend distributional spectral results on the algebra generated by Toeplitz sequences, by including the (pseudo) inversion operation, in the case where the sequences that are (pseudo) inverted are distributed as sparsely vanishing symbols. Applications to preconditioning and a potential use in image/signal restoration problems are presented
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