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

Journal Article · Thu Aug 31 04:00:00 EDT 2006 · Sbornik. Mathematics

DOI:https://doi.org/10.1070/SM2006V197N07ABEH003788; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)· OSTI ID:21266992

Radzievskii, G V ^[1]

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1. Institute of Mathematics of Ukrainian National Academy of Sciences, Kiev (Ukraine)

One considers the spectral problem x{sup (n)}+ Fx={lambda}x with boundary conditions U{sub j}(x)=0, j=1,...,n, for functions x on [0,1]. It is assumed that F is a linear bounded operator from the Hoelder space C{sup {gamma}}, {gamma} element of [0,n-1), into L{sub 1} and the U{sub j} are bounded linear functionals on C{sup k{sub j}} with k{sub j} element of {l_brace}0,...,n- 1{r_brace}. Let P{sub {zeta}} be the linear span of the root functions of the problem x{sup (n)}+ Fx={lambda}x, U{sub j}(x)=0, j=1,...,n, corresponding to the eigenvalues {lambda}{sub k} with |{lambda}{sub k}|<{zeta}{sup n}, and let E{sub {zeta}}(f){sub W{sub p}{sup l}}:=inf{l_brace}||f-g||{sub W{sub p}{sup l}}:g element of P{sub {zeta}}{r_brace}. An estimate of E{sub {zeta}}(f){sub W{sub p}{sup l}} is obtained in terms of the K-functional K({zeta}{sup -m},f;W{sub p}{sup l},W{sub p,U}{sup l+m}):= inf{l_brace}||f-x||{sub W{sub p}{sup l}}+{zeta}{sup -m}||x||{sub W{sub p}{sup l}{sup +}{sup m}}:x element of W{sub p}{sup l+m}, U{sub j}(x)=0 for k{sub j}<l+m{r_brace} (the direct theorem) and an estimate of this K-functional is obtained in terms of E{sub {xi}}(f){sub W{sub p}{sup l}} for {xi}{<=}{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{sup (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.

• OSTI ID: 21266992
• Journal Information: Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 7 Vol. 197; ISSN 1064-5616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
APPROXIMATIONS
BOUNDARY CONDITIONS
BOUNDARY-VALUE PROBLEMS
EIGENVALUES
FUNCTIONALS
MATHEMATICAL SPACE
PERIODICITY