Simultaneous rational approximation of a function and its derivatives in the complex plane (Q1070141)

Some aspects of simultaneous rational approximation of a function f(z) and its derivatives on the unit circle are investigated. The function f(z) is assumed to be analytic in some annulus containing the unit circle, and given a nonnegative integer \(\ell\), \(\| f\| =\max \{\| f^{(j)}\|_{p,1}:0\leq j\leq \ell \},\) where \(\| \cdot \|_{p,1}\) is the usual \(L_ p\) norm (1\(\leq p\leq \infty)\) on the unit circle. It is shown that the polynomial of simultaneous best approximation in the above norm, is just a polynomial of best approximation to \(f^{(\ell)}\), suitably integrated. Further, sharp asymptotic results are obtained for the case where the order of the derivative, namely \(\ell\), tends to infinity. For example, if f is meromorphic in \({\mathbb{C}}\) of finite order \(\rho\), with \(\nu\) poles in all, none lying on the unit circle, and if \(0\leq \mu <\min \{1,1/\rho \}\), then \[ \limsup_{m\to \infty}(\min_{R\in {\mathcal R}_{m\nu}}\max_{0\leq j\leq \mu m}\| f^{(j)}- R^{(j)}\|_{p,1})^{1/m \log m}=e^{\mu -1/\rho}. \]