This article is cited in 2 papers

On the matrix Fourier filtering problem for a class of models of nonlinear optical systems with a feedback

A. V. Razgulin, S. V. Sazonova

Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia

Abstract: A novel statement of the Fourier filtering problem based on the use of matrix Fourier filters instead of conventional multiplier filters is considered. The basic properties of the matrix Fourier filtering for the filters in the Hilbert-Schmidt class are established. It is proved that the solutions with a finite energy to the periodic initial boundary value problem for the quasi-linear functional differential diffusion equation with the matrix Fourier filtering Lipschitz continuously depend on the filter. The problem of optimal matrix Fourier filtering is formulated, and its solvability for various classes of matrix Fourier filters is proved. It is proved that the objective functional is differentiable with respect to the matrix Fourier filter, and the convergence of a version of the gradient projection method is also proved.

Key words: Fourier filtering, Hilbert–Schmidt matrix, functional differential diffusion equation, functional, gradient, models of nonlinear optical systems, feedback.

UDC: 519.651

Received: 26.05.2016

DOI: 10.7868/S0044466917090125

 English version: Computational Mathematics and Mathematical Physics, 2017, 57:9, 1385–1403

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