Uniform Rational Approximation of Even and Odd Continuations of Functions

The behavior of the best rational approximations of an odd continuation of a function is studied. It is shown that without additional conditions on the smoothness of the function, it is impossible to estimate the best rational approximation of the odd continuation of the function on $[-1,1]$ in terms of the best rational approximation of the original function on $[0,1]$. A sharp upper bound is found for the best rational approximations of an even (odd) continuation of a function in terms of an odd (even) continuation and an extremal Blaschke product.