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On strong unicity of $ L\sb{1}$-approximation


Author: András Kroó
Journal: Proc. Amer. Math. Soc. 83 (1981), 725-729
MSC: Primary 41A52
DOI: https://doi.org/10.1090/S0002-9939-1981-0630044-4
MathSciNet review: 630044
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Abstract: Let $ {C_1}$ be the space of continuous functions on $ [0,1]$ with norm $ \left\Vert f \right\Vert = \int_0^1 {\vert f(x)\vert} dx$, and let $ G \subset {C_1}$ be a finite dimensional unicity subspace, i.e. any $ f \in {C_1}$ possesses a unique best approximation out of $ G$. Consider an arbitrary $ f \in {C_1}$ such that 0 is its best approximation. Then for any $ 0 < \delta < {\delta _0}$ and $ \tilde g \in G$ with $ \left\Vert {f - \tilde g} \right\Vert \leqslant \left\Vert f \right\Vert + \delta $ it follows that $ \left\Vert {\tilde g} \right\Vert \leqslant Kw_f^* (\delta )$, where $ w_f^*(h) = {(w_f^{ - 1}(h)h)^{ - 1}}$ and $ {w_f}(h)$ denotes the modulus of continuity of $ f$. ($ - 1$ is used to denote the inverse function.)


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  • [1] B. O. Björnestal, Continuity of the metric projection operator. II, preprint, Dept. of Math., Royal Institute of Technology, Stockholm, TRITA-MAT, no. 20, 1974.
  • [2] -, Local Lipschitz continuity of the metric projection operator, Banach Center Publications, vol. 4, Approximation Theory, PWN, Warsaw, 1979, pp. 43-53. MR 553755 (80k:41019)
  • [3] M. P. Carroll and D. Braess, On uniqueness of $ {L_1}$-approximation for certain families of spline functions, J. Approx. Theory 12 (1974), 362-364. MR 0358150 (50:10615)
  • [4] E. W. Cheney and D. E. Wulbert, The existence and unicity of best approximations, Math. Scand. 24 (1969), 113-140. MR 0261241 (41:5857)
  • [5] P. V. Galkin, The uniqueness of the element of best mean approximation to a continuous function using splines with fixed knots, Math. Notes 15 (1974), 3-8. MR 0338623 (49:3387)
  • [6] A. Kroó, The continuity of best approximations in the space of integrable functions, Acta Math. Acad. Sci. Hungar. 32 (1978), 331-348. MR 512407 (81b:41067)
  • [7] -, Best $ {L_1}$-approximation on finite point sets: rate of convergence, J. Approx. Theory (to appear).
  • [8] D. J. Newman and H. S. Shapiro, Some theorems on Čebyšev approximation, Duke Math. J. 30 (1963), 673-681. MR 0156138 (27:6070)
  • [9] I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer, Berlin, Heidelberg and New York, 1970. MR 0270044 (42:4937)
  • [10] H. Strauss, $ {L_1}$-approximation mit Spline functionen, Numerical Methods in Approximation Theory, vol. 2 (L. Collatz and G. Meinardus, Eds.), Birkhäuserverlag, Stuttgart, 1975.
  • [11] K. H. Usow, On $ {L_1}$-approximation. II. Computation for discrete functions and discretization effects, SIAM J. 4 (1967), 233-244. MR 0217499 (36:588)
  • [12] R. Wegmann, Bounds for nearly best approximations, Proc. Amer. Math. Soc. 52 (1975), 252-256. MR 0442563 (56:944)

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DOI: https://doi.org/10.1090/S0002-9939-1981-0630044-4
Article copyright: © Copyright 1981 American Mathematical Society

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