Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Weak Chebyshev subspaces and continuous selections for the metric projection

Authors: Günther Nürnberger and Manfred Sommer
Journal: Trans. Amer. Math. Soc. 238 (1978), 129-138
MSC: Primary 41A50; Secondary 41A65
MathSciNet review: 482912
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let G be an n-dimensional subspace of $ C[a,b]$. It is shown that there exists a continuous selection for the metric projection if for each f in $ C[a,b]$ there exists exactly one alternation element $ {g_f}$, i.e., a best approximation for f such that for some $ a \leqslant {x_0} < \cdots < {x_n} \leqslant b$,

$\displaystyle \varepsilon {( - 1)^i}(f - {g_f})({x_i}) = \left\Vert {f - {g_f}} \right\Vert,\quad i = 0, \ldots ,n,\varepsilon = \pm 1.$

Further it is shown that this condition is fulfilled if and only if G is a weak Chebyshev subspace with the property that each g in G, $ g \ne 0$, has at most n distinct zeros. These results generalize in a certain sense results of Lazar, Morris and Wulbert for $ n = 1$ and Brown for $ n = 5$.

References [Enhancements On Off] (What's this?)

  • [1] A. L. Brown, On continuous selections for metric projections in spaces of continuous functions, J. Functional Analysis 8 (1971), 431–449. MR 0296666
  • [2] R. C. Jones and L. A. Karlovitz, Equioscillation under nonuniqueness in the approximation of continuous functions, J. Approximation Theory 3 (1970), 138–145. MR 0264303
  • [3] Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0204922
  • [4] A. J. Lazar, D. E. Wulbert, and P. D. Morris, Continuous selections for metric projections, J. Functional Analysis 3 (1969), 193–216. MR 0241952
  • [5] Günter Meinardus, Approximation of functions: Theory and numerical methods, Expanded translation of the German edition. Translated by Larry L. Schumaker. Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. MR 0217482
  • [6] G. Nürnberger, Dualität von Schnitten für die metrische Projektion und von Fortsetzungen kompakter Operatoren, Dissertation, Erlangen, 1975.
  • [7] Günther Nürnberger, Schnitte für die metrische Projektion, J. Approximation Theory 20 (1977), no. 2, 196–219 (German). MR 0470584
  • [8] P. Schwartz, Almost-Chebyshev subspaces of finite dimension in $ C(Q)$, (preprint).
  • [9] Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Translated from the Romanian by Radu Georgescu. Die Grundlehren der mathematischen Wissenschaften, Band 171, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin, 1970. MR 0270044
  • [10] Ivan Singer, The theory of best approximation and functional analysis, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 13. MR 0374771
  • [11] Manfred Sommer and Hans Strauss, Eigenschaften von schwach Tschebyscheffschen Räumen, J. Approximation Theory 21 (1977), no. 3, 257–268 (German). MR 0467119

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 41A50, 41A65

Retrieve articles in all journals with MSC: 41A50, 41A65

Additional Information

Keywords: Continuous selection, metric projection, weak Chebyshev spaces, alternation elements
Article copyright: © Copyright 1978 American Mathematical Society