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

Abstract

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 l, ∥ f∥= max{∥ f ( j) ∥ p,1 :0 ⩽ j ⩽ l}, where ∥·∥ p,1 is the usual L p norm (1 ⩽ p ⩽ ∞) 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 ( l) , suitably integrated. Further, sharp asymptotic results are obtained for the case where the order of the derivative, namely l, tends to infinity. For example, if f is meromorphic in C of finite order ϱ, with v poles in all, none lying on the unit circle, and if 0 ⩽ μ < min{1, 1 ϱ } , then ▪.