Uniform approximation through partitioning

Abstract

In this paper, the problem of best uniform polynomial approximation to a continuous function on a compact set X X is approached through the partitioning of X X and the definition of norms corresponding to the partition and each of the standard L p {L_p} norms 1 ≦ p > ∞ 1 \leqq p > \infty . For computational convenience, a pseudo norm is defined corresponding to each partition. When the partition is chosen appropriately, the corresponding best approximations (using both the norms and the pseudo norm) are arbitrarily close to a best uniform approximation. A chracterization theorem for best pseudo norm approximation is presented, along with an alternation theorem for best pseudo norm approximation to a univariate function.