Strong unicity criterion in some space of operators

Grzegorz Lewicki

Commentationes Mathematicae Universitatis Carolinae (1993)

  • Volume: 34, Issue: 1, page 81-87
  • ISSN: 0010-2628

Abstract

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Let X be a finite dimensional Banach space and let Y ⊂ X be a hyperplane. Let L Y = { L ∈ L ( X , Y ) : L ∣ Y = 0 } . In this note, we present sufficient and necessary conditions on L 0 ∈ L Y being a strongly unique best approximation for given L ∈ L ( X ) . Next we apply this characterization to the case of X = l ∞ n and to generalization of Theorem I.1.3 from [12] (see also [13]).

How to cite

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Lewicki, Grzegorz. "Strong unicity criterion in some space of operators." Commentationes Mathematicae Universitatis Carolinae 34.1 (1993): 81-87. <http://eudml.org/doc/247466>.

@article{Lewicki1993,
abstract = {Let $X$ be a finite dimensional Banach space and let $Y\subset X$ be a hyperplane. Let $\text\{L\}\,_Y=\lbrace L\in \text\{L\}\,(X,Y):L\mid _Y=0\rbrace $. In this note, we present sufficient and necessary conditions on $L_0\in \text\{L\}\,_Y$ being a strongly unique best approximation for given $L\in \text\{L\}\,(X)$. Next we apply this characterization to the case of $X=l_\infty ^n$ and to generalization of Theorem I.1.3 from [12] (see also [13]).},
author = {Lewicki, Grzegorz},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {best approximation; strongly unique best approximation; approximation in spaces of linear operators; strongly unique best approximation},
language = {eng},
number = {1},
pages = {81-87},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Strong unicity criterion in some space of operators},
url = {http://eudml.org/doc/247466},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Lewicki, Grzegorz
TI - Strong unicity criterion in some space of operators
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 1
SP - 81
EP - 87
AB - Let $X$ be a finite dimensional Banach space and let $Y\subset X$ be a hyperplane. Let $\text{L}\,_Y=\lbrace L\in \text{L}\,(X,Y):L\mid _Y=0\rbrace $. In this note, we present sufficient and necessary conditions on $L_0\in \text{L}\,_Y$ being a strongly unique best approximation for given $L\in \text{L}\,(X)$. Next we apply this characterization to the case of $X=l_\infty ^n$ and to generalization of Theorem I.1.3 from [12] (see also [13]).
LA - eng
KW - best approximation; strongly unique best approximation; approximation in spaces of linear operators; strongly unique best approximation
UR - http://eudml.org/doc/247466
ER -

References

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  3. Bartelt M.W., McLaughlin H.W., Characterizations of strong unicity in approximation theory, Jour. Approx. Th. 9 (1973), 255-266. (1973) Zbl0273.41019MR0372500
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  6. Cheney E.W., Introduction to Approximation Theory, McGraw-Hill, New York, 1966. Zbl0912.41001MR0222517
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  9. Laurent P.J., Approximation et Optimisation, Hermann, Paris, 1972. Zbl0238.90058MR0467080
  10. Lewicki G., Kolgomorov's type criteria for spaces of compact operators, Jour. Approx. Th. 64 (1991), 181-202. (1991) MR1091468
  11. Newman D.J., Shapiro H.S., Some theorems on Chebyshev approximations, Duke Math. J. 30 (1963), 673-681. (1963) MR0156138
  12. Odyniec Wł., Lewicki G., Minimal Projections in Banach Spaces, Lecture Notes in Math., vol. 1449 (1990), Springer-Verlag. Zbl1062.46500MR1079547
  13. Odyniec Wł., The uniqueness of minimal projection (in Russian), Izv. Vyss. Ucebn. Zavied. Matematika 3 (1978), 73-75. (1978) MR0626041
  14. Smarzewski R., Strongly unique best approximation in Banach spaces, Jour. Approx. Th. 47 (1986), 184-194. (1986) Zbl0615.41027MR0847538
  15. Sudolski J., Wȯjcik A., Some remarks on strong uniqueness of best approximation, Approx. Theory and its Appl. 6 (2), June 1990, 44-78. (2) MR1078687

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