Nonlinear tensor product approximation of functions

We are interested in approximation of a multivariate function f(x1,…,xd)f(x1,…,xd) by linear combinations of products u1(x1)⋯ud(xd)u1(x1)⋯ud(xd) of univariate functions ui(xi)ui(xi), i=1,…,di=1,…,d. In the case d=2d=2 it is the classical problem of bilinear approximation. In the case of approximation in the L2L2 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)f(x1,x2). There are known results on the rate of decay of errors of best bilinear approximation in LpLp under different smoothness assumptions on ff. The problem of multilinear approximation (nonlinear tensor product approximation) in the case d≥3d≥3 is more difficult and much less studied than the bilinear approximation problem. We will present results on best multilinear approximation in LpLp under mixed smoothness assumption on ff.