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

For the given data (wi,xi,yi)(wi,xi,yi), i=1,…,Mi=1,…,M, we consider the problem of existence of the best discrete approximation in lplp norm (1≤p<∞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 lplp 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.