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 Mat. Sb., 2006, Volume 197, Number 7, Pages 87–136 (Mi msb1117)

Direct and inverse theorems on approximation by root functions of a regular boundary-value problem

Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: One considers the spectral problem $x^{(n)}+Fx=\lambda x$ with boundary conditions $U_j(x)=0$, $j=1,…,n$, for functions $x$ on $[0,1]$. It is assumed that $F$ is a linear bounded operator from the Hölder space $C^\gamma$, $\gamma\in[0,n-1)$, into $L_1$ and the $U_j$ are bounded linear functionals on $C^{k_j}$ with $k_j\in\{0,…,n-1\}$. Let $\mathfrak P_\zeta$ be the linear span of the root functions of the problem $x^{(n)}+Fx=\lambda x$, $U_j(x)=0$, $j=1,…,n$, corresponding to the eigenvalues $\lambda_k$ with $|\lambda_k|<\zeta^n$, and let $\mathscr E_\zeta(f)_{W_p^l}:=\inf\{\|f-g\|_{W_p^l}:g\in\mathfrak P_\zeta\}$. An estimate of $\mathscr E_\zeta(f)_{W_p^l}$ is obtained in terms of the $K$-functional
$K(\zeta^{-m},f;W_p^l,W_{p,U}^{l+m}) :=\inf\{\|f-x\|_{W_p^l} +\zeta^{-m}\|x\|_{W_p^{l+m}}: x\in W_p^{l+m}, U_j(x)=0 for k_j<l+m\}$
(the direct theorem) and an estimate of this $K$-functional is obtained in terms of $\mathscr E_\xi(f)_{W_p^l}$ for $\xi\leqslant\zeta$ (the inverse theorem).
In several cases two-sided bounds of the $K$-functional are found in terms of appropriate moduli of continuity, and then the direct and the inverse theorems are stated in terms of moduli of continuity. For the spectral problem $x^{(n)}=\lambda x$ with periodic boundary conditions these results coincide with Jackson's and Bernstein's direct and inverse theorems on the approximation of functions by a trigonometric system.
Bibliography: 41 titles.

DOI: https://doi.org/10.4213/sm1117

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English version:
Sbornik: Mathematics, 2006, 197:7, 1037–1083

Bibliographic databases:

UDC: 517.927.6+517.518
MSC: 41A17, 34L20

Citation: G. V. Radzievskii, “Direct and inverse theorems on approximation by root functions of a regular boundary-value problem”, Mat. Sb., 197:7 (2006), 87–136; Sb. Math., 197:7 (2006), 1037–1083

Citation in format AMSBIB
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