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A $ 2$-parameter Chebyshev set which is not a sun

Author: Charles B. Dunham
Journal: Proc. Amer. Math. Soc. 50 (1975), 315-316
MSC: Primary 41A65
MathSciNet review: 0402369
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Abstract: Consider approximation with respect to the Chebyshev norm $ \vert\vert g\vert\vert = \sup \{ \vert g(x)\vert:0 \leq x \leq 1\} $ on $ [0,1]$. A subset $ G$ of $ C[0,1]$ such that each $ f \in C[0,1]$ has a unique best approximation from $ G$ is called a Chebyshev set. It has been shown by the author that there exist Chebyshev sets which are not suns [2], but the examples given were essentially one-dimensional. An example is now given which is two-dimensional.

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  • [1] C. B. Dunham, Existence and continuity of the Chebyshev operator, SIAM Rev. 10 (1968), 444–446. MR 0238011,
  • [2] -, Chebyshev sets in $ C[0,1]$ which are not suns, Canad. Math. Bull. (to appear).
  • [3] 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

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