| Title:
|
Strong unicity criterion in some space of operators (English) |
| Author:
|
Lewicki, Grzegorz |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
1 |
| Year:
|
1993 |
| Pages:
|
81-87 |
| . |
| Category:
|
math |
| . |
| Summary:
|
\font\muj=rsfs10 \font\mmuj=rsfs8 Let $X$ be a finite dimensional Banach space and let $Y\subset X$ be a hyperplane. Let $\text{\mmuj L}\,_Y=\{L\in \text{\mmuj L}\,(X,Y):L\mid _Y=0\}$. In this note, we present sufficient and necessary conditions on $L_0\in \text{\mmuj L}\,_Y$ being a strongly unique best approximation for given $L\in \text{\mmuj L}\,(X)$. Next we apply this characterization to the case of $X=l_\infty ^n$ and to generalization of \linebreak Theorem I.1.3 from [12] (see also [13]). (English) |
| Keyword:
|
best approximation |
| Keyword:
|
strongly unique best approximation |
| Keyword:
|
approximation in spaces of linear operators |
| MSC:
|
41A35 |
| MSC:
|
41A50 |
| MSC:
|
41A52 |
| MSC:
|
41A65 |
| MSC:
|
46B99 |
| idZBL:
|
Zbl 0785.41023 |
| idMR:
|
MR1240206 |
| . |
| Date available:
|
2009-01-08T18:01:18Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118558 |
| . |
| Reference:
|
[1] Baronti M., Franchetti C.: Minimal and polar projections onto hyperplanes in the space $l^p$ and $l_\infty $.to appear. |
| Reference:
|
[2] Baronti M., Papini P.L.: Norm one projections onto subspaces of $l^p$.Ann. Mat. Pura Appl. IV (1988), 53-61. MR 0980971 |
| Reference:
|
[3] Bartelt M.W., McLaughlin H.W.: Characterizations of strong unicity in approximation theory.Jour. Approx. Th. 9 (1973), 255-266. Zbl 0273.41019, MR 0372500 |
| Reference:
|
[4] Bartelt M.W., Schmidt D.P.: On Poreda's problem on the strong unicity constants.Jour. Approx. Th. 33 (1981), 69-79. Zbl 0478.41024, MR 0639222 |
| Reference:
|
[5] Blatter J., Cheney E.W.: Minimal projections onto hyperplanes in sequence spaces.Ann. Mat. Pura Appl. 101 (1974), 215-227. MR 0358179 |
| Reference:
|
[6] Cheney E.W.: Introduction to Approximation Theory.McGraw-Hill, New York, 1966. Zbl 0912.41001, MR 0222517 |
| Reference:
|
[7] Franchetti C.: Projections onto hyperplanes in Banach spaces.Jour. Approx. Th. 38 (1983), 319-333. Zbl 0516.41028, MR 0711458 |
| Reference:
|
[8] Chalmers B.L., Metcalf F.T., Taylor G.D.: Strong unicity of arbitrary rate.Jour. Approx. Th. 37 (1983), 238-245. Zbl 0517.41031, MR 0693010 |
| Reference:
|
[9] Laurent P.J.: Approximation et Optimisation.Hermann, Paris, 1972. Zbl 0238.90058, MR 0467080 |
| Reference:
|
[10] Lewicki G.: Kolgomorov's type criteria for spaces of compact operators.Jour. Approx. Th. 64 (1991), 181-202. MR 1091468 |
| Reference:
|
[11] Newman D.J., Shapiro H.S.: Some theorems on Chebyshev approximations.Duke Math. J. 30 (1963), 673-681. MR 0156138 |
| Reference:
|
[12] Odyniec Wł., Lewicki G.: Minimal Projections in Banach Spaces.Lecture Notes in Math., vol. 1449 (1990), Springer-Verlag. Zbl 1062.46500, MR 1079547 |
| Reference:
|
[13] Odyniec Wł.: The uniqueness of minimal projection (in Russian).Izv. Vyss. Ucebn. Zavied. Matematika 3 (1978), 73-75. MR 0626041 |
| Reference:
|
[14] Smarzewski R.: Strongly unique best approximation in Banach spaces.Jour. Approx. Th. 47 (1986), 184-194. Zbl 0615.41027, MR 0847538 |
| Reference:
|
[15] Sudolski J., Wȯjcik A.: Some remarks on strong uniqueness of best approximation.Approx. Theory and its Appl. 6 (2), June 1990, 44-78. MR 1078687 |
| . |
