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 Mat. Sb., 1998, Volume 189, Number 4, Pages 83–124 (Mi msb312)

Direct and converse theorems in problems of approximation by vectors of finite degree

Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: Let $A$ be a linear operator in a complex Banach space $X$ with domain $\mathfrak D(A)$ and a non-empty resolvent set. An element $g\in \mathfrak D_\infty (A):=\bigcap _{j=0,1,…\mathfrak D(A^j)$ is called a vector of degree at most $\zeta (>0)$ with respect to $A$ if $\|A^jg\|_X\leqslant c(g)\zeta ^j$, $j=0,1,…$ . The set of vectors of degree at most $\zeta$ is denoted by $\mathfrak G_\zeta (A)$. The quantity $E_\zeta (f,A)_X=\inf _{g\in \mathfrak G_\zeta (A)}\|f-g\|_X$ is introduced and estimated in terms of the $K$-functional $K(\zeta ^{-r},f;X,\mathfrak D(A^r)) =\inf _{g\in \mathfrak D(A^r)}(\|f-g\|_X+\zeta ^{-r}\|A^rf\|_X)$ (the direct theorem). An estimate of this $K$-functional in terms of $E_\zeta (f,A)_X$ and $\|f\|_X$ is established (the converse theorem). Using the estimates obtained, necessary and sufficient conditions for the following properties are found in terms of $E_\zeta (f,A)_X$: 1) $f\in \mathfrak D_\infty (A)$; 2) the series $e^{zA}f:=\sum _{r=0}^\infty (z^rA^rf)/(r!)$ converges in some disc; 3) the series $e^{zA}f$ converges in the entire complex plane. The growth order and the type of the entire function $e^{zA}f$ are calculated in terms of $E_\zeta (f,A)_X$.

DOI: https://doi.org/10.4213/sm312

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English version:
Sbornik: Mathematics, 1998, 189:4, 561–601

Bibliographic databases:

UDC: 517.43+517.5
MSC: 41A65, 41A17

Citation: G. V. Radzievskii, “Direct and converse theorems in problems of approximation by vectors of finite degree”, Mat. Sb., 189:4 (1998), 83–124; Sb. Math., 189:4 (1998), 561–601

Citation in format AMSBIB
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This publication is cited in the following articles:
1. G. V. Radzievskii, “Characterization of Hadamard vector classes in terms of least deviations of their elements from vectors of finite degree”, Sb. Math., 192:12 (2001), 1829–1876
2. M. L. Gorbachuk, “Growth Order of an Operator Exponential on Entire Vectors”, Funct. Anal. Appl., 36:1 (2002), 62–64
3. Radzievskii, GV, “Direct and inverse theorems for least deviations from the root functions of regular boundary value problems”, Doklady Mathematics, 71:1 (2005), 35
4. G. V. Radzievskii, “Direct and inverse theorems on approximation by root functions of a regular boundary-value problem”, Sb. Math., 197:7 (2006), 1037–1083
5. Grushka Ya., Torba S., “Direct and Inverse Theorems in the Theory of Approximation of Banach Space Vectors by Exponential Type Entire Vectors”, Modern Analysis and Applications: Mark Krein Centenary Conference, Operator Theory Advances and Applications, 190, 2009, 295–314
6. M. І. Dmytryshyn, “Tensor Products of Approximation Spaces Associated with Regular Elliptic Operators”, J Math Sci, 2015
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