Nonexistence of continuous selections of the metric projection for a class of weak Chebyshev spaces
Author:
Manfred Sommer
Journal:
Trans. Amer. Math. Soc. 260 (1980), 403409
MSC:
Primary 41A65; Secondary 41A50, 41A52
MathSciNet review:
574787
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Abstract: The class of all those ndimensional weak Chebyshev subspaces of whose elements have no zero intervals is considered. It is shown that there does not exist any continuous selection of the metric projection for G if there is a nonzero g in G having at least distinct zeros. Together with a recent result of NürnbergerSommer, the following characterization of continuous selections for is valid: There exists a continuous selection of the metric projection for G in if and only if each nonzero g in G has at most n distinct zeros.
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 S. Karlin and W. J. Studden, Tchebycheff systems: With applications in analysis and statistics, Interscience, New York, 1966. MR 0204922 (34:4757)
 [3]
 A. J. Lazar, P. D. Morris and D. E. Wulbert, Continuous selections for metric projections, J. Functional Analysis 3 (1969), 193216. MR 0241952 (39:3288)
 [4]
 G. Nürnberger and M. Sommer, Weak Chebyshev subspaces and continuous selections for the metric projection, Trans. Amer. Math. Soc. 238 (1978), 129138. MR 482912 (80a:41027)
 [5]
 , Characterization of continuous selections of the metric projection for spline functions, J. Approximation Theory 22 (1978), 320330. MR 0481769 (58:1868)
 [6]
 G. Nürnberger, Continuous selections for the metric projection and alternation, J. Approximation Theory (to appear).
 [7]
 M. Sommer, Continuous selections of the metric projection for 1Chebyshev spaces, J. Approximation Theory 26 (1978), 4653. MR 536714 (80h:41018)
 [8]
 , Characterization of continuous selections for the metric projection for generalized splines, SIAM J. Math. Anal. 11 (1980), 2340. MR 556494 (81a:41038)
 [9]
 B. Stockenberg, On the number of zeros of functions in a weak Tchebyshevspace, Math. Z. 156 (1977), 4957. MR 0487208 (58:6868)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198005747873
PII:
S 00029947(1980)05747873
Keywords:
Continuous selection,
metric projection,
weak Chebyshev spaces
Article copyright:
© Copyright 1980 American Mathematical Society
