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On proximinality in $ L\sb{1}(T\times S)$

Authors: S. M. Holland, W. A. Light and L. J. Sulley
Journal: Proc. Amer. Math. Soc. 86 (1982), 279-282
MSC: Primary 41A65; Secondary 41A44, 41A50
MathSciNet review: 667289
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Abstract: It is proved that if $ G$ and $ H$ are finite-dimensional subspaces of $ {L_1}(S)$ and $ {L_1}(T)$ respectively then each element of $ {L_1}(T \times S)$ has a best approximation in the subspace $ {L_1}(T) \otimes G + H \otimes {L_1}(S)$.

References [Enhancements On Off] (What's this?)

  • [1] E. W. Cheney, J. H. McCabe, W. A. Light and G. Phillips, The approximation of bivariate functions by sums of univariate ones using the $ {L_1}$-metric, Center for Numerical Analysis Technical Report, University of Texas, 1979.
  • [2] N. Dunford and J. T. Schwartz, Linear operators, Part 1, Interscience, New York, 1959.
  • [3] W. A. Light and E. W. Cheney, Some best-approximation theorems in tensor-product spaces, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 3, 385–390. MR 602291,
  • [4] J. R. Respess Jr. and E. W. Cheney, Best approximation problems in tensor-product spaces, Pacific J. Math. 102 (1982), no. 2, 437–446. MR 686562

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Article copyright: © Copyright 1982 American Mathematical Society

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