Uniform approximation through partitioning

Abstract:

In this paper, the problem of best uniform polynomial approximation to a continuous function on a compact set $X$ is approached through the partitioning of $X$ and the definition of norms corresponding to the partition and each of the standard ${L_p}$ norms $1 \leqq p < \infty$. 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.