This is an interesting question.
In addition to the incisive comments by @Peter Breuer, there is the issue of the type of product of two functions to approximate, e.g., dot product, inner product, direct product and tensor product. A very good study of direct products in given in Section 2.3.3, starting on page 14, in
http://www.kinetics.nsc.ru/chichinin/books/spectroscopy/Bartok_Partay10.pdf
Tensor products viewed as functions on a manifold are approximated by the Kronecker product of two SU(N) matrices in
http://arxiv.org/pdf/1206.7060.pdf
As regards the question for a product as the corresponding question for a sum, I just would like to remark that one can map the first to the second by taking logarithms, the second to the first by exponentiation. - Besides thinking in error norms, one can also try to tackle the problem in terms of function theory (analyticity assumptions), power series or asymptotic expansions. Also, orthogonal expansions may be a point of view, that may be related to Hilbert spaces. One could also tackle the question by algebra methods like formal power series, that is, without regard to convergence. Using power series and/or asymptotic expansions, one then has an approximation by truncation of the series/expansion. One may also use something like the tau method of Lanzcos. - As regards the question which type of product, one may use convolution of functions here. Then, there is a mapping by Fourier transformation from multiplication of two functions to convolution of their transforms and vice versa . - A further interesting question might be, in what sense the product of two distributions (that is not defined easily if at all) may be approximated by a further distribution, and in what sense. - In view of all these possibilities, I strongly suspect, that there are probably as many answer as there are possibilities or even more. Thus, one gains quite a lot by studying specific examples.
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