Exact n-Widths of Integral Operators

Abstract

Let $K(x,y) \in C([0,1]) \times [0,1])$ and set

$${K_p} = \left\{ {K\,h\,\left( x \right):K\,h\,\left( x \right) = \int\limits_0^1 {K\,(x,y)\,h\,(y)\,d\,y,{{\left\| h \right\|}_p}} \underline{\underline < } \,1} \right\}$$

where

$${\left\| h \right\|_p} = \left\{ {_{ess\sup \,\left\{ {\left| {h\left( y \right)} \right|:\,0\underline{\underline < } \,y\,\underline{\underline < } \,1} \right\},\,\,\,\,\,p = \infty }^{{{\left( {\int\limits_0^1 {{{\left| {h\left( y \right)} \right|}^p}dy} } \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$p$}}}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\underline{\underline < } p < \infty }} \right.$$

In Section 2 of this chapter we determine the n-widths (d _n , dⁿ and δ _n ) of K _p in L^q for p = ∞, 1 ≦ q ≦ ∞, and 1 ≦ p ≦ ∞, q = 1, where K is a nondegenerate totally positive kernel. (Such kernels were defined in Section 5 of Chapter IV and are redefined in Section 2.) We prove that all three n-widths considered are equal (the common value depending on p and q) and that there exists a set of n distinct points in {E_i} ⁿ_i=1 in (0,1) for which X _n = span {K(•,E_i)} ⁿ_i=1 is optimal for d _n (K _p ;L^q), with the {E _i } ⁿ_i=1 dependent on p, q. Furthermore, there exists an additional set {η _i } ⁿ_i=1 of n distinct points in (0,1) (again dependent on p, q) for which interpolation from X _n to K h(x) at the {η _i } ⁿ_i=1 is optimal for δ _n (K _p ;L^q). (An analogous statement holds for dⁿ(K _p ;L^q).)