Uniform rational approximation of the odd and even Cauchy transforms

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 $x^{\alpha}$, $x\in[0,1]$, to $[-1,1]$ for all $alpha\in(0,+\infty)\setminus(2\mathbb N-1)$, 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 $alpha\in(0,+\infty)\setminus\mathbb N$ the best rational approximations of the even and odd extensions of the above function show the same weak asymptotic behaviour. Bibliography: 29 titles.