Strong coapproximation in Banach spaces
Let X be a normed linear space and M a non-empty proper subset of X. An element \(m\in M\) is a best coapproximation to \(x\in X\) if \(\| m-y\| \leq \| x-y\|\) for each \(y\in M\). If \(g: [0,\infty]\to [0,\infty]\) is an increasing convex function and \(g(0)=0\) then \(m\in M\) is a strong coapproximation, relative to g, to \(x\in X\) if, for some \(c=c(x)>0\), \(g(\| m-y\| \leq g(\| x-y\|)-cg(\| x-m\|)\) for all \(y\in M\). The set M is a cosun if whenever m is a best coapproximation to x it is also a best coapproximation to \(m+t(x-m)\) for each \(t>0\). The principal and representative result is that if M is a cosun in one of the spaces \(L_ p(S,\Sigma,\mu)\), the Hardy spaces \(H^ p\) or the Sobolev spaces \(H^{k,p}(T)\) (T open in \({\mathbb{R}}^ n)\) for \(1<p<\infty\) then for \(q=\max (2,p)\). \(g(t)=t^ q\), each best coapproximation \(m\in H\) to \(x\in X\) is a strong coapproximation with constant c independent of \(x\in X\). Explicit values of c are given in these cases.
scientific article; zbMATH DE number 4105543
scientific article