Arithmetic of semigroups of series in multiplicative systems |
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Authors: | I P Il’inskaya |
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Institution: | 1.Kharkov National University,Kharkov,Ukraine |
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Abstract: | We study the arithmetic of a semigroup MP\mathcal{M}_{\mathcal{P}} of functions with operation of multiplication representable in the form f(x) = ?n = 0¥ ancn(x) ( an 3 0,?n = 0¥ an = 1 ) f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} , where { cn }n = 0¥ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup MP\mathcal{M}_{\mathcal{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 I0(MP)I_0(\mathcal{M}_{\mathcal{P}}) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that
is dense in MP\mathcal{M}_{\mathcal{P}} in the topology of uniform convergence. |
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Keywords: | |
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