Some problems in the approximation of functions of two variables and n-widths of integral operators

Abstract

We explicitly obtain, for K(x, y) totally positive, a best choice of functions u1, ..., un and v1, ..., vn for the problem minui, vi (∝01 (∝01 |K(x, y) - ∑i = 1, n ui(x) vi(y)| dyp dx) 1 p, where ui ε{lunate} Lp[0, 1], vi ε{lunate} L1[0, 1], i = 1, ..., n, and p ε{lunate} [1, ∞]. We show that an optimal choice is determined by certain sections K(x, ξ1), ..., K(x, ξn), and K(τ1, y), ..., K(τn, y) of the kernel K. We also determine the n-widths, both in the sense of Kolmogorov and of Gel'fand, and identify optimal subspaces, for the set Kr,v = {f:f(x) = ∑ i=1 raiki(x) + ∫ 0 1K(x,y)h(y)dy, (a1, ..., ar)ε{lunate}Rr, {norm of matrix}h{norm of matrix}p≤1}, as a subset of Lq[0, 1], with either p = ∞ and q ε{lunate} [1, ∞], or p ε{lunate} [1, ∞] and q = 1, where {k1(x), ..., kr(x), K(x, y)} satisfy certain restrictions. A particular example is the ball Br,v = {f:fr-1} in the Sobolev space. © 1978.