Bögel Functions, Tensor Products and Blending Approximation

For two compact metric spaces X and Y, we prove that the space of all bounded Bogel functions on X x Y is isometrically isomorphic with the completion of the blending function space M ( X ) 0 C ( Y ) + C ( X )@ M ( Y ) with respect to a suitable norm. This solves a problem raised by G. FREUD.Here M ( Z ) and C ( Z )denote the spaces of all bounded and continuous functions on the metric space Z , respectively. Applications are given to some problems in so-called blending approximation. All this is based on an abstract Jackson-type theorem in terms of the entropy numbers of X and Y and the mixed modulus of smoothness.