
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 steadystate 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 matrixvalued positive selfadjoint secondorder 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.
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Mathematics of the USSRSbornik, 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. USSRSb., 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 147173
\mathnet{http://mi.mathnet.ru/msb1991}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=732383}
\zmath{https://zbmath.org/?q=an:0568.350060542.35038}
\transl
\jour Math. USSRSb.
\yr 1985
\vol 51
\issue 1
\pages 141167
\crossref{https://doi.org/10.1070/SM1985v051n01ABEH002852}
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V. S. Stepanov, “The attractor of the equation of oscillations of a thin elastic rod”, Russian Math. Surveys, 40:3 (1985), 245–246

Babin A., “The Belonging of DifferentialEquation Solutions to Nikolsky Spaces”, 289, no. 6, 1986, 1289–1293

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

A. V. Babin, “On the smoothness of solutions of differential equations at singular points of the boundary of the domain”, Math. USSRIzv., 37:3 (1991), 489–510

Gorodetskii V., “Polynomial Representation of Solutions of OperatorDifferential Equations of Hyperbolic Type in HilbertSpace”, Differ. Equ., 27:6 (1991), 656–660

Kulakov A., “Convergence of Weighted Polynomial Approximations to Solutions of Partial Differential Equations with Quasianalytical Coefficients”, J. Approx. Theory, 93:3 (1998), 458–479

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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