Best Approximation by Smooth Functions and Related Problems

SATTES (1980) has considered best uniform approximation by smooth functions in the space C([−1,1]) of continuous real valued functions on an interval and has characterised such best approximations. GLASHOFF (1980) and PINKUS (1980) have considered a family of similar problems. We state a general problem which contains those of Sattes, Glashoff and Pinkus as special cases. A general theorem which characterises best approximations is proved in Section 2 and in Section 4 it is used to obtain an alternative proof of Sattes’ result. The link between the general theorem and the proof of Sattes’ result is provided by a theorem, concerning the zeros of certain functions, which is proved in Section 3. Section 3 is a small contribution to the theory of total positivity.