Nonlinear tensor product approximation of functions

We are interested in approximation of a multivariate function f(x1, . . . , xd) by linear combinations of products u (x1) · · ·u(xd) of univariate functions u(xi), i = 1, . . . , d. In the case d = 2 it is the classical problem of bilinear approximation. In the case of approximation in the L2 space the bilinear approximation problem is closely related to the problem of singular value decomposition (also called Schmidt expansion) of the corresponding integral operator with the kernel f(x1, x2). There are known results on the rate of decay of errors of best bilinear approximation in Lp under different smoothness assumptions on f . The problem of multilinear approximation (nonlinear tensor product approximation) in the case d ≥ 3 is more difficult and much less studied than the bilinear approximation problem. We will present results on best multilinear approximation in Lp under mixed smoothness assumption on f .