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Proceedings of the American Mathematical Society

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Bounds for nearly best approximations


Author: Rudolf Wegmann
Journal: Proc. Amer. Math. Soc. 52 (1975), 252-256
MSC: Primary 41A65
DOI: https://doi.org/10.1090/S0002-9939-1975-0442563-1
MathSciNet review: 0442563
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Abstract: Let $ X$ be a uniformly convex space and $ \psi $ be the inverse function of the modulus of convexity $ \delta ( \cdot )$. Assume here that $ \psi $ is a concave function. Let $ V$ be a linear subspace of $ X$ and let $ f$ in $ X$ be such that $ \vert\vert f\vert\vert = 1 = \min \{ \vert\vert f - v\vert\vert:v\epsilon V\} $. Then for $ 0 < \delta \leqslant 1$ and for $ v$ in $ V$ with $ \vert\vert f - v\vert\vert \leqslant 1 + \delta $, it follows that $ \vert\vert v\vert\vert \leqslant K \cdot \psi (\delta )$. Let $ T$ be a compact Hausdorff-space and $ V$ a finite-dimensional subspace of $ C(T,X)$. When $ V$ has the interpolation property $ ({P_m})$ with $ V = m \cdot \dim X$, then the same type of estimate as above holds.


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

  • [1] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396-414. MR 1501880
  • [2] Ky Fan and I. Glicksberg, Some geometric properties of the spheres in a normed linear space, Duke Math. J. 25 (1958), 553-568. MR 20 #5421. MR 0098976 (20:5421)
  • [3] G. Freud, Eine Ungleichung für Tschebyscheffsche Approximations polynome, Acta Sci. Math. (Szeged) 19 (1958), 162-164. MR 21 #251. MR 0101440 (21:251)
  • [4] S. J. Poreda, On the continuity of best polynomial approximations, Proc. Amer. Math. Soc. 36 (1972), 471-476. MR 0316717 (47:5264)
  • [5] A. Schönhage, Approximationstheorie, de Gruyter, Berlin, 1971. MR 43 #3693. MR 0277960 (43:3693)
  • [6] I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Editura Academiei Republicii Socialiste România, Bucharest, 1967; English transl., Die Grundlehren der math. Wissenschaften, Band 171, Springer-Verlag, Berlin and New York, 1970. MR 38 #3677; 42 #4937. MR 0270044 (42:4937)

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DOI: https://doi.org/10.1090/S0002-9939-1975-0442563-1
Article copyright: © Copyright 1975 American Mathematical Society

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