On the existence of the best discrete approximation in lp norm by reciprocals of real polynomials

AbstractFor the given data (wi,xi,yi), i=1,…,M, we consider the problem of existence of the best discrete approximation in lp 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 lp 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