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. In this article, a sufficient condition for the vector index absolute summation method was obtained in terms of the best approximation by «angle» of the functions from Lebesgue space. The theorem that gives a sufficient condition proves the conditions that are sufficient in different cases, which may depend on the parameters. From this proved theorem, a sufficient condition on the term mixed smoothness modulus of the function from Lebesgue space, which is easily obtained by a well-known inequality, is also presented.