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

This article is cited in 6 scientific papers (total in 6 papers)

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

G. V. Radzievskii

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
Received: 06.05.1997

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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\by G.~V.~Radzievskii
\paper Direct and converse theorems in problems of approximation by vectors of finite degree
\jour Mat. Sb.
\yr 1998
\vol 189
\issue 4
\pages 83--124
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\crossref{https://doi.org/10.4213/sm312}
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\transl
\jour Sb. Math.
\yr 1998
\vol 189
\issue 4
\pages 561--601
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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. M. L. Gorbachuk, “Growth Order of an Operator Exponential on Entire Vectors”, Funct. Anal. Appl., 36:1 (2002), 62–64  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathscinet  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathscinet  zmath  isi
    6. M. І. Dmytryshyn, “Tensor Products of Approximation Spaces Associated with Regular Elliptic Operators”, J Math Sci, 2015  crossref  scopus  scopus  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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