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 Mat. Sb. (N.S.), 1984, Volume 123(165), Number 2, Pages 147–173 (Mi msb1991)

This article is cited in 7 scientific papers (total in 7 papers)

Construction and investigation of solutions of differential equations by methods in the theory of approximation of functions

A. V. Babin

Abstract: The steady-state equation $Au_0=f$, the parabolic Cauchy problem $u_1'(t)=Au_1(t)$, $u_1(0)=f$, and the hyperbolic problem $u_2"(t)=Au_2(t)$, $u_2(0)=f$, $u_2'(0)=0$, are considered, where $A$ is a matrix-valued positive selfadjoint second-order partial differential operator with analytic coefficients, and $f$ is an analytic function.
Methods in the theory of weighted approximation of functions by polynomials on the line are used to construct polynomial representations of solutions of these problems of the form $u_i=\lim_{h\to\infty}P_n^i(A)f$, where the polynomials $P_n^i(\lambda)$, $i=0,1,2$, are constructed in explicit form. Estimates of the rate of convergence are given. With the help of these estimates and Bernstein's inverse theorems in approximation theory, theorems are obtained on the smoothness and analyticity of solutions of degenerate systems whose coefficients are trigonometric polynomials.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1985, 51:1, 141–167

Bibliographic databases:

UDC: 517.944
MSC: Primary 35A35, 35B65, 41A10; Secondary 35A10, 35A30, 35C99, 41A17, 41A25, 42A10
Received: 10.11.1982

Citation: A. V. Babin, “Construction and investigation of solutions of differential equations by methods in the theory of approximation of functions”, Mat. Sb. (N.S.), 123(165):2 (1984), 147–173; Math. USSR-Sb., 51:1 (1985), 141–167

Citation in format AMSBIB
\Bibitem{Bab84} \by A.~V.~Babin \paper Construction and investigation of solutions of differential equations by methods in the theory of approximation of functions \jour Mat. Sb. (N.S.) \yr 1984 \vol 123(165) \issue 2 \pages 147--173 \mathnet{http://mi.mathnet.ru/msb1991} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=732383} \zmath{https://zbmath.org/?q=an:0568.35006|0542.35038} \transl \jour Math. USSR-Sb. \yr 1985 \vol 51 \issue 1 \pages 141--167 \crossref{https://doi.org/10.1070/SM1985v051n01ABEH002852} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. S. Stepanov, “The attractor of the equation of oscillations of a thin elastic rod”, Russian Math. Surveys, 40:3 (1985), 245–246
2. Babin A., “The Belonging of Differential-Equation Solutions to Nikolsky Spaces”, 289, no. 6, 1986, 1289–1293
3. A. V. Babin, “Connection between analytic properties of operator functions and smoothness of solutions of degenerate differential equations”, Funct. Anal. Appl., 22:1 (1988), 48–50
4. A. V. Babin, “On the smoothness of solutions of differential equations at singular points of the boundary of the domain”, Math. USSR-Izv., 37:3 (1991), 489–510
5. Gorodetskii V., “Polynomial Representation of Solutions of Operator-Differential Equations of Hyperbolic Type in Hilbert-Space”, Differ. Equ., 27:6 (1991), 656–660
6. Kulakov A., “Convergence of Weighted Polynomial Approximations to Solutions of Partial Differential Equations with Quasianalytical Coefficients”, J. Approx. Theory, 93:3 (1998), 458–479
7. Zelik, S, “Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities”, Discrete and Continuous Dynamical Systems, 11:2–3 (2004), 351
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