On norms in weighted spaces of functions given on infinite intervals (Q1103821)

Let f be a function on \((1,\infty)\) that has absolutely continuous derivative of order r, \(\phi\) a non-negative measurable function. Suppose that \(\| \phi f^{(r)}\|_{L^ p}<\infty\) and \(\| t^{\alpha}\phi^{-1}\|_{L^ q}<\infty\), where \(\alpha >r-1\), \(1/p+1/q=1\). Then it is proved that there exists a unique polynomial \(P=\sum^{r-1}_{k=0}a_ kt\) k such that \[ \lim_{t\to \infty}(f(t)- p(t))^{(m)}=0,\quad m=0,1,...,r-1. \] Moreover P satisfies the inequality \[ \| (f-P)^{(m)} t^{\alpha -r+m}\|_{L^ 1} \leq \frac{1}{\alpha (\alpha -1)...(\alpha -r+m+1)}\| \phi f^{(r)}\|_ P. \]