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Some best-approximation theorems in tensor-product spaces

  • W. A. Light (a1) and E. W. Cheney (a1)

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We begin by describing a concrete example from the class of problems to be considered. A continuous bivariate function f defined on the square |t| ≤ 1, |s| ≤ 1 is to be approximated by a tensor-product form involving univariate functions. For example, the approximation may be prescribed to have the form

in which the Ti are the Tchebycheff polynomials, and the coefficient functions xi(t) and yi(s) are to be chosen to achieve a good or best approximation. Will a best approximation exist? If so, how can it be obtained?

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References

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(1)Diliberto, S. P. and Straus, E. G.On the approximation of a function of several variables by sums of functions of fewer variables, Pacific J. Math. 1 (1951), 195210.
(2)Dunford, N. and Schwartz, J. T.Linear Operators. Part I (Interscience, New York, 1959).
(3)Havinson, S.YA. A Cebyshev theorem for the approximation of a function of two variables by sums π(x) + ψ(y), Izv. Akad. Nauk SSSR 33 (1969, 650–666. MR41 no. 7351. English translation, Math. USSR-Izv. 3 (1969), 617632.
(4)Kuratowski, K. and Ryll-Nardzewski, C.A general theorem on selectors, Bull. Acad. Polon. Sci. Sir. Sci. Math. Astronom. Phys. 13 (1965), 397403. MR32 no. 6421.
(5)Light, W. A. and Cheney, E. W. On the approximation of a bivariate function by sums of univariate ones, J. Approximation Theory, to appear. Center for Numerical Analysis, Report 140, The University of Texas, Austin, August 1978.
(6)Respess, J. The theory of best approximation in tensor product spaces, Ph.D. Dissertation, The University of Texas at Austin, 1980.
(7)Rudin, W.Real and Complex Analysis (McGraw-Hill, New York, 1966).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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