Abstract
In [DD] the problem of existence and uniqueness of a quadrature formula (QF) with maximal trigonometric degree of precision (MTDP) with a fixed number of free nodes and fixed different multiplicities at each node is considered. Even the affirmative answer to the question of existence and uniqueness is useless from a practical point of view if the QF is not explicitly found or if a complete characterization for the nodes and for the coefficients of the QF is not given. On the other hand the problem of the complete constructive characterization of the QF with MTDP is one of the main problems in the theory of numerical integration. In this paper we give a complete constructive characterization for the QF with MTDP in the case of a special type of periodic multiplicities. The results can be considered as a natural generalization of the previous results, which are given in [GO] (one-periodic case of multiplicities) and [DD] (two-periodic case of multiplicities). We evaluate the practical usefulness of the optimal numerical methods, which are obtained.
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