Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T00:56:17.387Z Has data issue: false hasContentIssue false
  • English
  • Français

The L1-version of the Diliberto–Straus algorithm in C(T × S)

Published online by Cambridge University Press:  20 January 2009

W. A. Light  and
S. M. Holland
W. A. Light
Affiliation:
Mathematics Department, Texas A. ' M. University, College Station, Texas
S. M. Holland
Affiliation:
Mathematics Department, Texas A. ' M. University, College Station, Texas
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let (S, Σ, μ) and (T, Θ, v) be two measure spaces of finite measure where we assume S, T are compact Hausdorff spaces and μ, v are regular Borel measures. We construct the product measure space (T x S, >, Φ σ) in the usual way. Let G = [gl, g2, …, gp] and H = [hl, h1, …, hm be finite dimensional subspaces of C(S) and C(T) respectivelywhere G and H are also Chebyshev with respect to the L1-norm. Note that a subspace Y of a normed linear space X is Chebyshev if each x ∈X possesses exactly one best approximation y ∈Y. For example, in C(S) with the L1-norm, the subspace of polynomials of degree at most n is a Chebyshev subspace. This is an old theorem of Jackson. Now set

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 27 , Issue 1 , February 1984 , pp. 31 - 45
Copyright
Copyright © Edinburgh Mathematical Society 1984

References

REFERENCES

1.Deutsch, F., The alternating method of von Neumann, Multivariate Approximation Theory, eds. Schempp, W. and Zeller, K. (Birchhäuser Verlag, Basel, 1979).Google Scholar
2.Diliberto, S. P. and Straus, E. G., On the approximation of a function of several variables by the sum of functions of fewer variables, Pacific J. Math. 1 (1951), 195–210.CrossRefGoogle Scholar
3.Dunford, N. and Schwartz, J. T., Linear Operators, Part J (Interscience, New York, 1959).Google Scholar
4.Holland, S. M., Light, W. A. and Sulley, L. J., On proximinality in L1(TxS), Proc. Amer. Math. Soc, to appear.Google Scholar
5.Holmes, R. B., A Course on Optimization and Best Approximation (Springer-Verlag, 1972).CrossRefGoogle Scholar
6.James, R. C., Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265–292.CrossRefGoogle Scholar
7.Light, W. A. and Cheney, E. W., Some best approximation theorems in Tensor Product Spaces, Math. Proc. Camb. Phil. Soc. 8 (1981), 385–390.CrossRefGoogle Scholar
8.Light, W. A., The Diliberto-Straus algorithm in L 1 (X x Y), J. Approx. Th., to appear.Google Scholar
9.Light, W. A., Mccabe, J. H., Phillips, G. H. and Cheney, E. W., The approximation of bivariate functions by sums of univariate ones using the L 1-metric, Proc. Edinburgh Math. Soc. 25 (1982), 173–181.CrossRefGoogle Scholar
10.Singer, I., Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces (Springer-Verlag, New York, Heidelberg, Berlin, 1970).CrossRefGoogle Scholar
11.Von Neumann, J., Functional Operators Part II, The geometry of orthogonal spaces (Annals of Math. Studies, no. 22, Princeton University Press, 1950).Google Scholar