No abstract

Let "W = {u> t (0)£i 0 t * t n e classical system of the Walsh functions, «5V the multiplicative semigroup of the functions represented by series of functions w^ (t) with non-negative coefficients which sum equals 1. We study the arithmetic of 5f-#/. The analogues of the well-known Khinchin factorization theorems related to the arithmetic of the convolution semigroup of probability measures on the real line are valid in 5^iti-The classes of idempotent elements, of infinitely divisible elements, of elements without indecomposable factors, and of elements without indecomposable and non-degenerate idempotent factors are completely described. We study also the class of indecomposable elements. Our method is based on the following fact: S^tf is isomorphic to the semigroup of probability measures on the group of characters of the Cantor-Walsh group.2000 Mathematics subject classification: primary 60B15,43A25; secondary 42C10.

Il'inskaya UDC 519.21 We study the arithmetic of a semigroup M P of functions with operation of multiplication representable in the form f x a x n n n ( ) ( ) = = ∞ ∑ χ 0 a a n n n ≥ = = ∞ ∑ ( ) 0 1 0 , , where { } χ n n= ∞ 0 is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup M P , analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R n are true. We describe the class I 0 ( ) M P of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in M P in the topology of uniform convergence.

Il'inskaya. Phase retrieval for probability measures on cyclic groups, Mat. Stud. 46 (2016), 89-95.We construct the classes of probability measures on the groups Z n and Z for which the identification of a measure given modulus of its characteristic function is possible up to a shift and the central symmetry. For groups Z 3 , Z 4 we give the completely description of such classes.