Approximation of a two-variate function by linear combinations of its crosses

It is known (Mazur, Grothendieck, Enflo, Davie) that not every function $f(x,y)$ continuous on the square can uniformly be approximated by linear combinations of its crosses $f(a,y)f(x,b)$. We obtain various conditions on a continuous two-variate function $f$ ensuring that it can uniformly be approximated by linear combinations of its crosses. We prove that these conditions are sufficient using so-called maximal cross approximations, defined by analogy with one special type of cross approximations of matrices. Unlike the uniform case, cross approximation in $L_2$ is always possible, and we discuss various algorithms for this approximation. We also present examples where consecutive cross approximations of continuous functions diverge in $L_p$. Bibliography: 23 titles.