On nonuniqueness in rational \(L_ p\)-approximation (Q1103138)

Using topological arguments, the author proves that each \(m+2\) dimensional subspace E of \(L_ p[a,b]\) \((1<p<\infty\), \(a<b)\) contains a function having at least two best approximations from R m (rational functions of numerator degree \(\leq m\), denominator degree \(\leq n)\) under the single condition \(E\cap R\) \(m_ n=\{0\}\). In a note added in proof it is pointed out that the same arguments lead to the existence of an element in C[a,b] for which the best uniform approximation from \(R\) \(m_ n\) is degenerate.