Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-14T04:45:36.774Z Has data issue: false hasContentIssue false
  • English
  • Français

Some best-approximation theorems in tensor-product spaces

Published online by Cambridge University Press:  24 October 2008

W. A. Light  and
E. W. Cheney
W. A. Light
Affiliation:
University of Lancaster and University of Texas at Austin
E. W. Cheney
Affiliation:
University of Lancaster and University of Texas at Austin
Get access Rights & Permissions [Opens in a new window]

Extract

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?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 89 , Issue 3 , May 1981 , pp. 385 - 390
Copyright
Copyright © Cambridge Philosophical Society 1981

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(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), 195–210.CrossRefGoogle Scholar
(2)Dunford, N. and Schwartz, J. T.Linear Operators. Part I (Interscience, New York, 1959).Google Scholar
(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), 617–632.CrossRefGoogle Scholar
(4)Kuratowski, K. and Ryll-Nardzewski, C.A general theorem on selectors, Bull. Acad. Polon. Sci. Sir. Sci. Math. Astronom. Phys. 13 (1965), 397–403. MR32 no. 6421.Google Scholar
(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.Google Scholar
(6)Respess, J. The theory of best approximation in tensor product spaces, Ph.D. Dissertation, The University of Texas at Austin, 1980.Google Scholar
(7)Rudin, W.Real and Complex Analysis (McGraw-Hill, New York, 1966).Google Scholar