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A note on proximinality in C(S × T) with the L1-norm

Published online by Cambridge University Press:  20 January 2009

W. A. Light
W. A. Light
Affiliation:
Department of Mathematics, University of Lancaster, Lancaster, LA1 4YL
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Let S and T be compact Hausdorff spaces and G and H finite-dimensional subspaces of C(S) and C(T) respectively. Suppose μ and ν are regular Borel measures on S and T respectively such that μ(S)= ν(T)= 1. The product measure μ × ν will be denoted by σ. Set U = G⊗C(T), V =C(S)⊗H and W = U + V. If G and H possess continuous proximity maps, then U and V are proximinal subspaces of C(S × T) when this linear space is equipped with the L1-norm, [4, Lemma 2]. That is, every z∈C(S × T) possesses at least one best approximation from U and from V. A metric selection Au:C(S × T →U is a mapping which associates each z ∈ C(S × T) with one of its best approximations in U.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 29 , Issue 3 , October 1986 , pp. 335 - 341
Copyright
Copyright © Edinburgh Mathematical Society 1986

References

REFERENCES

1.Holmes, R. B., A Course on Optimisation and Best Approximation (Springer Verlag, 1972).Google Scholar
2.Jackson, D., A note on a class of polynomials of approximation, Trans. Amer. Math. Soc. 22 (1921), 320–326.CrossRefGoogle Scholar
3.Light, W. A. and Cheney, E. W., Some best approximation theorems in Tensor-Product Spaces, Math. Proc. Camb. Philos. Soc. 89 (1981), 385–390.CrossRefGoogle Scholar
4.Light, W. A. and Holland, S. M., The L1-version of the Diliberto-Straus algorithm in C(TxS), Proc. Edinburgh Math. Soc. 27 (1984), 31–45.CrossRefGoogle Scholar
5.Light, W. A., McCabe, J. H., Phillips, G. M. and Cheney, E. W., The approximation of bivariate functions by sums of univariate ones using the L1-metric, Proc. Edinburgh Math. Soc. 25 (1982), 173–181.Google Scholar
6.Respess, J. R. Jr. and Cheney, E. W., Best approximation problems in Tensor-Product Spaces, Pacific J. Math. 102 (1982), 437–446.Google Scholar