The relationship between the zeros of best approximations and differentiability

Abstract

We examine the relationship between the analytic properties of continuous functions on [ − 1 , 1 ] [ - 1,1] and the location of the roots of the sequence of best polynomial approximations. We show that if the approximants have no zeros in a certain ellipse then the function being approximated must be analytic in this ellipse. We also show that the rate at which the zeros of the n n th approximant tend to the interval [ − 1 , 1 ] [ - 1,1] determines the global differentiability of the function under consideration.