The best recovery of the solution of the Dirichlet problem from inaccurate spectrum of the boundary function

E. V. Abramova

Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 4, pp. 3-12

Résumé

In many applied problems appears a situation where it is necessary to recover the value of a function from some information (usually not exact or complete). The general problem of the optimal recovery of a linear functional on a class of functions from finite information first appeared in the works of S. A. Smolyak. In the future this subject has received a fairly wide development in a variety of ways. There are many approaches which solve similar problems. Here we follow the approach that assumes the existence of a priori information about the object whose characteristics are to be recovered. This allows us to set the problem of finding the best method for recovering this characteristic among all possible recovery methods. This view of reconstruction tasks ideologically goes back to Kolmogorov's work in the years 1930s on finding the best means of approximation for classes of functions. The mathematical theory, where recovery problems are studied on the basis of this approach, has been actively developing in recent decades, revealing close links with the classical problems of approximation theory and having various applications to the problems of practice. This paper is devoted to the problem of best recovery of the solution of the Dirichlet problem in the $L_2$ metric on the line in the upper half-plane parallel to the abscissa axis, according to the following information about the boundary function: the boundary function belongs to some Sobolev space of functions, and its Fourier transform knows an approximate (in the $L_\infty$ metric) on finite segment symmetric with respect to zero. An optimal recovery method is constructed and the exact value of the optimum recovery error is found. It should be noted that the optimal method uses, generally speaking, not all available information, and the one that uses it, in a certain way, “smoothes out”.

Bibliographie

[1] Smolyak S. A., On Optimal Reconstruction of Functions and Functionals of Them, Kand. diss., MGU, M., 1965 (in Russian)

[2] Micchelli C. A., Rivlin T. J., “A survey of optimal recovery”, Optimal Estimation in Approximation Theory, eds. C. A. Micchelli, T. J. Rivlin, Plenum Press, N. Y., 1977, 1–54 | MR

[3] Melkman A. A., Micchelli C. A., “Optimal estimation of linear operators in Hilbert spaces from inaccurate data”, SIAM J. Numer. Anal., 16 (1979), 87–105 | DOI | MR | Zbl

[4] Micchelli C. A., Rivlin T. J., “Lectures on Optimal Recovery”, Lect. Notes Math., 1129, Springer-Verlag, Berlin, 1985, 21–93 | DOI | MR

[5] Magaril-Il'yaev G. G., Osipenko K. Y., “Optimal recovery of functions and their derivatives from inaccurate information about the spectrum and inequalities for derivatives”, Functional Analysis and Its Applications, 37:3 (2003), 51–64 (in Russian) | DOI

[6] Magaril-Il'yaev G. G., Osipenko K. Y., “On the Reconstruction of Convolution-type operators from inaccurate information”, Proc. Steklov Instit. Math., 269 (2010), 174–185 | DOI | MR | Zbl

[7] Magaril-Il'yaev G. G., Osipenko K. Y., “On Optimal Harmonic Synthesis from Inaccurate Spectral Data”, Functional Analysis and Its Applications, 44:3 (2010), 76–79 (in Russian) | DOI | Zbl

[8] Stein E., Weiss G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971 | MR | Zbl

[9] Magaril-Il'yaev G. G., Tikhomirov V. M., Convex Analysis and its Applications, Editorial URSS, M., 2011 (in Russian)

[10] Kolmogorov A. N., Fomin S. V., Elements of the Theory of Functions and Functional Analysis, Nauka, M., 1976 (in Russian)