Sums of products of functions of x times functions of y

AbstractWe study functions u of the form (∗) u(x, y) = Σnk=1Ak(x)Bk(y), mostly by means of two functional determinants, Wn(u) and Dn(u). A sample result is that if u has the form (∗) on an open subset of a region G on which u is real analytic, then u must have this form (with possibly different Ak and Bk) on all of G. We also study the representability of u(x, y) in the form (∗), on a region G in which u is either n times differentiable or is merely continuous. The question whether the limit of functions of the form (∗) is again of the form (∗) is treated. Moreover, we study those functions of the form (∗) that are solutions of some classical partial differential equations