Uniform continuity of generalized rational approximations
In a real Banach space \(X\), let \(\rho(x, A)\) be the distance from \(x \in X\) to a subset \(A \subset X\). A mapping \(\Phi: X \to A\) is called a (multiplicative) \(\varepsilon\)-selection if for each \(x \in X\) one has \[ \|\Phi(x)-x\|\leq (1+\varepsilon) \rho(x,A). \] An \(\varepsilon\)-selection obviously exists for every \(\varepsilon>0\) but it is not necessarily a continuous mapping. In the paper under review, \(X=C[0,1]\) and \(A\) is defined by means of two fixed subspaces \(V\) and \(W\) as the set \(R_{V,W}\) of all continuous functions \(f\) of the form \(f=v/w\), \(v \in V\), \(w \in W\). The author proves that under some mild assumptions concerning \(V\) and \(W\) for any \(\varepsilon \in (0,2)\) there is no uniformly continuous \(\varepsilon\)-selection from a ball in \(C[0,1]\) to \(R_{V,W}\). This fact is valid, in particular, when \(V\) and \(W\) are the subspaces of algebraic polynomials of degrees \(m\) and \(n\), respectively; the case \(m=0\), \(n=1\) has been established earlier by Konyagin. For \(\varepsilon \in (0,1)\), the above results can be extended to the case of the complex space \(C[0,1]\).
scientific article; zbMATH DE number 1928166