Direct and inverse theorems on approximation by root

G. V. Radzievskii

Sbornik. Mathematics, Tome 197 (2006) no. 7, pp. 1037-1083

Résumé

One considers the spectral problem $x^{(n)}+Fx=\lambda x$ with boundary conditions $U_j(x)=0$, $j=1,\dots,n$, 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 $k_j\in\{0,\dots,n-1\}$. Let $\mathfrak P_\zeta$ be the linear span of the root functions of the problem $x^{(n)}+Fx=\lambda x$, $U_j(x)=0$, $j=1,\dots,n$, corresponding to the eigenvalues $\lambda_k$ with $|\lambda_k|<\zeta^n$, and let $\mathscr E_\zeta(f)_{W_p^l}:=\inf\bigl\{\|f-g\|_{W_p^l}:g\in\mathfrak P_\zeta\bigr\}$. An estimate of $\mathscr E_\zeta(f)_{W_p^l}$ is obtained in terms of the $K$-functional $K(\zeta^{-m},f;W_p^l,W_{p,U}^{l+m}) :=\inf\bigl\{\|f-x\|_{W_p^l} +\zeta^{-m}\|x\|_{W_p^{l+m}}: x\in W_p^{l+m},\ U_j(x)=0\text{ for }k_j<l+m\bigr\}$ (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.

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