Vol. 102, No. 2, 1982

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Best approximation problems in tensor-product spaces

J. R. Respess and Elliott Ward Cheney, Jr.

Vol. 102 (1982), No. 2, 437–446

This paper concerns an existence problem for best approximations of bivariate functions. The approximating functions are taken from infinite-dimensional subspaces having tensor product form. Problems of this type arise, for example, in approximating the kernel of an integral equation by a degenerate (“separable”) kernel. A sample of our results is this: let G and H be finite-dimensional subspaces in continuous function spaces C(S) and C(T) respectively. If one of these subspaces has a continuous proximity map and the other a Lipschitzian proximity map, then G C(T) + C(S) H is proximinal in C(S × T); i.e., best approximations exist in this subspace.

Mathematical Subject Classification 2000
Primary: 41A65
Secondary: 41A45, 46E10, 46M05
Received: 18 June 1981
Published: 1 October 1982
J. R. Respess
Elliott Ward Cheney, Jr.