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Direct and inverse theorems on approximation by root
functions of a regular boundary-value problem
G. V. Radzievskii Institute of Mathematics, Ukrainian National Academy of Sciences
Abstract:
One considers the spectral problem
$x^{(n)}+Fx=\lambda x$ with boundary conditions
$U_j(x)=0$,
$j=1,\dots,n$,
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.
UDC:
517.927.6+
517.518
MSC: 41A17,
34L20 Received: 27.06.2005
DOI:
10.4213/sm1117
Fulltext:
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