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Uniform approximation through partitioning


Author: S. E. Weinstein
Journal: Math. Comp. 26 (1972), 493-503
MSC: Primary 41A50
DOI: https://doi.org/10.1090/S0025-5718-1972-0308666-2
MathSciNet review: 0308666
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Abstract: In this paper, the problem of best uniform polynomial approximation to a continuous function on a compact set $ X$ is approached through the partitioning of $ X$ and the definition of norms corresponding to the partition and each of the standard $ {L_p}$ norms $ 1 \leqq p < \infty $. For computational convenience, a pseudo norm is defined corresponding to each partition. When the partition is chosen appropriately, the corresponding best approximations (using both the norms and the pseudo norm) are arbitrarily close to a best uniform approximation. A chracterization theorem for best pseudo norm approximation is presented, along with an alternation theorem for best pseudo norm approximation to a univariate function.


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

  • [1] E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, New York, 1966. MR 36 #5568 MR 0222517 (36:5568)
  • [2] R. H. Harris, Uniform Approximation of Functions: Approximation by Partitioning, Thesis, University of Utah, Salt Lake City, Utah, 1970.
  • [3] S. E. Weinstein, ``Approximations of functions of several variables: Product Chebychev approximations. I,'' J. Approximation Theory, v. 2, 1969, pp. 433-447. MR 40 #7683. MR 0254475 (40:7683)
  • [4] S. E. Weinstein, ``Uniform approximation of functions through optimal partitioning,'' SIAM J. Numer. Anal. (To appear.) MR 0402364 (53:6185)
  • [5] S. E. Weinstein, ``Computation of best uniform approximations through partitioning.'' (In preparation.)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1972-0308666-2
Keywords: Approximation, uniform approximation, multivariate approximation
Article copyright: © Copyright 1972 American Mathematical Society

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