Approximations of Two-Place Functions by Finite Sums of Terms with Separable Variables Original Research Article

It is known that if a smooth function h in two real variables x and y belongs to the class Σn of all sums of the form Σnk=1ƒk(x) gk(y), then its (n + 1)th order "Wronskian" det[hxiyj]ni,j=0 is identically equal to zero. The present paper deals with the approximation problem h(x, y) ≍ Σnk=1ƒk(x) gk(y) with a prescribed n, for general smooth functions h not lying in Σn. Two natural approximation sums T=T(h) ∈ Σn, S=S(h) ∈ Σn are introduced and the errors |h-T|, |h-S| are estimated by means of the above mentioned Wronskian of the function h. The proofs utilize the technique of ordinary linear differential equations.