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Finite dimensional sigmaPi-approximations and data compression. (English) Zbl 0866.68122

Summary: The problem of \(\Sigma\Pi\)-approximation in the simplest form is the following: let \(f(x,y)\) be the real function of two real variables \(x\) and \(y\); we want to replace this function by the finite sum of products of the one-variable functions \(\sum^S_{k=1}\Phi_k(x)\Psi_k(y)\) and to provide some given accuracy of approximation. This problem is important in various applications, like data compression in digital image processing, in decomposition of two-dimensional digital filters into one-dimensional filters and so on. At the beginning of our century E. Schmidt (1907) considered this problem in the analytic form and found a connection between optimal \(\Sigma\Pi\)-approximation and singular values of the integral operator with the kernel \(f(x,y)\). After that many mathematicians were interested in this problem but usually in the analytic form without numerical algorithms. In this paper, we consider so-called finite-dimensional \(\Sigma\Pi\)-approximations in the general form and in the examples, and give the numerical algorithms for them.

MSC:

68U10 Computing methodologies for image processing
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