Sharper results

The results in this chapter have intrinsic interest and importance. But, additionally, this chapter is a sort of departure point or, say, a bridge to further multidimensional generalizations. Most of the earlier obtained results have been generalized to the multivariate case; various theorems of that kind can be found in the survey paper [128]. It is also summarized in [92] and in [122]. All these generalizations fail in sharpness, since too many terms fall into a trap of the remainder values. In other words, one-dimensional results are involved in multivariate extensions mostly as the remainder terms; even their leading terms contribute to the multidimensional leading terms too seldom.