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Uniform approximation through partitioning

Author: S. E. Weinstein
Journal: Math. Comp. 26 (1972), 493-503
MSC: Primary 41A50
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 Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
  • [2] R. H. Harris, Uniform Approximation of Functions: Approximation by Partitioning, Thesis, University of Utah, Salt Lake City, Utah, 1970.
  • [3] Stanley E. Weinstein, Approximations of functions of several variables: Product Chebychev approximations. I, J. Approximation Theory 2 (1969), 433–447. MR 0254475
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  • [5] S. E. Weinstein, ``Computation of best uniform approximations through partitioning.'' (In preparation.)

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Keywords: Approximation, uniform approximation, multivariate approximation
Article copyright: © Copyright 1972 American Mathematical Society

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