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.
