2019
Том 71
№ 4

# Il’inskaya I. P.

Articles: 2
Brief Communications (Russian)

### Probability Measures on the Group of Walsh Functions With Trivial Equivalence Class

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 717–721

We establish necessary and sufficient conditions for the retrieval, to within a shift, of a composition of three Poisson distributions and a uniform distribution on five or six elements of the group of Walsh functions according to the absolute values of their characteristic functions.

Article (Russian)

### Arithmetic of semigroups of series in multiplicative systems

Ukr. Mat. Zh. - 2009. - 61, № 7. - pp. 939–947

We study the arithmetic of a semigroup $\mathcal{M}_P$ of functions with operation of multiplication representable in the form $f(x)=∑^{∞}_{n=0} a_nχ_n(x)\left(a_n≥0,\; ∑^{∞}_{n=0}a_n =1 \right)$, where $\{χ_n|\}^{∞}_{n=0}$ is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup $\mathcal{M}_P$ , 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 $I_0(\mathcal{M}_P)$ of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in $\mathcal{M}_P$ in the topology of uniform convergence.