Some properties of the sum of the moduli of the terms of a grouped trigonometric series

For a trigonometric series grouped in a certain way and such that the coefficients are monotonic in each of the groups, the absolute convergence and properties of the sums of moduli of the terms thereof are examined. Necessary and sufficient conditions for this sum to lie in L-2 pi(p) - spaces, p is an element of [1, infinity], are obtained, and upper and lower bounds in terms for the coefficients of the series are established.