n-Widths in Approximation Theory pp 138-197 | Cite as
Exact n-Widths of Integral Operators
Chapter
Abstract
Let \(K(x,y) \in C([0,1]) \times [0,1])\) and set
where
In Section 2 of this chapter we determine the n-widths (d n , d n and δ n ) of K p in Lq 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 {Ei} i=1 n in (0,1) for which X n = span {K(•,Ei)} i=1 n is optimal for d n (K p ;L q ), with the {E i } i=1 n dependent on p, q. Furthermore, there exists an additional set {η i } i=1 n 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 } i=1 n is optimal for δ n
(K p ;L q ). (An analogous statement holds for d n (K p ;L q ).)
$${K_p} = \left\{ {K\,h\,\left( x \right):K\,h\,\left( x \right) = \int\limits_0^1 {K\,(x,y)\,h\,(y)\,d\,y,{{\left\| h \right\|}_p}} \underline{\underline < } \,1} \right\}$$
$${\left\| h \right\|_p} = \left\{ {_{ess\sup \,\left\{ {\left| {h\left( y \right)} \right|:\,0\underline{\underline < } \,y\,\underline{\underline < } \,1} \right\},\,\,\,\,\,p = \infty }^{{{\left( {\int\limits_0^1 {{{\left| {h\left( y \right)} \right|}^p}dy} } \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/
{\vphantom {1 p}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{$p$}}}},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\underline{\underline < } p < \infty }} \right.$$
Keywords
Integral Operator Periodic Function Extremal Problem Distinct Zero Optimal Rank
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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© Springer-Verlag Berlin Heidelberg 1985