On some nonlinear spaces of approximating functions (Q801492)

Let F be continuous and strictly positive on [0,\(\infty)\). Let \(V_ n\) denote the (non-linear) space of functions of the form F(\(\alpha\) x)P(x) where \(\alpha\geq 0\) and P is \(\pi_ n\). Various conditions on F are given to insure that \(V_ n\) is an existence space for C[a,b], \(0\leq a<b<\infty:\) i.e. so that every f in C[a,b] has a best approximation in \(V_ n\). In the simplest case the condition is that \(\lim_{x\to \infty}F(x)\) be finite and positive. In another result, the condition is that, for each \(x>1\), \(\lim_{\beta \to \infty}F(\beta x)/F(\beta)=\infty\). Another result involves positive rational functions. All proofs involve considering the best approximation in the space \(V_ n(\alpha)\), i.e. where \(\alpha\) is fixed, and then estimating the coefficients in the polynomial involved. Various examples are given to show that not much improvement in the hypotheses is possible. There are some remarks about the lack of uniqueness when \(F(x)=e^ x\), but these are not new.