Uniform rational approximation of the odd and even Cauchy transforms

Abstract

Best uniform rational approximations of the odd and even Cauchy transforms are considered. The results obtained form a basis for finding the weak asymptotics of best uniform rational approximations of the odd extension of the function x α , x ∈ [0, 1], to [−1, 1] for all α ∈ (0, +∞) \ (2N − 1), which complements some results due to Vyacheslavov. The strong asymptotics of the best rational approximations of this function on [0, 1] and its even extension to [−1, 1] were found by Stahl. It follows from these results that for α ∈ (0, +∞)\N the best rational approximations of the even and odd extensions of the above function show the same weak asymptotic behaviour.