On the existence of the best discrete approximation inlpnorm by reciprocals of real polynomials

Abstract

For the given data (w i , x i , y i ), i = 1, . . . , M, we consider the problem of existence of the best discrete approximation in l p norm (1 ≤ p < ∞) by reciprocals of real polynomials. For this problem, the existence of best approximations is not always guaranteed. In this paper, we give a condition on data which is necessary and sufficient for the existence of the best approximation in l p norm. This condition is theoretical in nature. We apply it to obtain several other existence theorems very useful in practice. Some illustrative examples are also included.