Let W n be the set of 2?-periodic functions with absolutely continuous (n&1)th derivatives and nth derivatives with essential suprema bounded by one. Let n>1.
Best uniform approximations to a periodic continuous function from W n are characterized. The result depends upon an analysis of the relation between the zeros, knots, and signs of periodic splines with simple knots. An appendix by O. V. Davydov states an alternative characterisation and demonstrates that the two characterisations are equivalent.
1998 Academic Press
1. Introduction
Throughout this paper ``periodic'' will mean periodic with period 2?. Let C be the space of periodic continuous real valued functions defined on the real line R, and let C be equipped with the uniform norm. For each n # N let W n be the set of those functions u # C which have absolutely continuous (n&1)th derivative u (n&1) , and whose nth derivatives satisfy the condition &u (n) & 1. The paper presents a characterization (Theorem 5.2.1), when n>1, of those u # W n which are best approximations from W n to a given v 0 # C .
Let C([0, 1]) be the space of continuous real valued functions defined on the interval [0,1], equipped with the uniform norm, and let W n be the set of those functions u # C([0, 1]) which have absolutely continuous (n&1)th derivatives, and whose nth derivatives satisfy the condition &u Korneichuk in 1961 [11] gave a characterization of best approximations u # W 1 to a given to v 0 # C([0, 1]). Sattes [13] gave a characterization for the cases n>1.