Abstract
co A system X = {X k (X)}k= 0 of complex-valued functions that are defined on the segment [0, i] is said to be multiplicative if for any two functions X k and Xs in it the functions (Xk)-I and XkXE also belong to it. The classical examples of multiplicative systems are the trigonometric system {exp " (2~inX)}n=_ ~ and the Walsh system. In [I-3] the properties of periodic multiplicative systems and, in particular, of certain lacunary series with respect to these systems have been investigated. For the trigonometric system, the most widely known and important lacunarity of a series is the Hadamard--lacunarity, i.e., the condition
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