Abstract
Abstract This article is devoted to the topic of absolute summation of series or Cesaro summation. The relevance of this article lies in the fact that a type of absolute summation with vector index, which has not been previously studied, is considered. The necessary condition of absolute summation with vector index of double trigonometric series is obtained and several sufficient conditions have been proved. In order to prove this necessary condition, the lemma providing the necessary condition of absolute summation with vector index of the double number series is first time proved. Another lemma, which is auxiliary to the main result, has been proven for the special case of absolute summation with vector index. With the help of these two lemmas the necessary condition is obtained. The theorem that gives a sufficient condition proves the conditions that are sufficient in different cases, which may depend on the parameters.