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

Abstract

AbstractWe 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)1p, where ui ϵ Lp[0, 1], vi ϵ L1[0, 1], i = 1, …, n, and p ϵ [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=1raiki(x) + ∫01K(x,y)h(y)dy, (a1, ..., ar)ϵRr, ‖h‖p⩽1}, as a subset of Lq[0, 1], with either p = ∞ and q ϵ [1, ∞], or p ϵ [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