Arithmetic of semigroups of series in multiplicative systems

  • I. P. Il’inskaya

We study the arithmetic of a semigroup \(\mathcal{M}_{\mathcal{P}}\) of functions with operation of multiplication representable in the form \( f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} \), where \( \left\{ {{\chi_n}} \right\}_{n = 0}^\infty \) is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup \(\mathcal{M}_{\mathcal{P}}\), 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 \(I_0(\mathcal{M}_{\mathcal{P}})\) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in \(\mathcal{M}_{\mathcal{P}}\) in the topology of uniform convergence.


Probability Measure Multiplicative Function Compact Abelian Group Walsh Function Multiplicative System 
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Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • I. P. Il’inskaya
    • 1
  1. 1.Kharkov National UniversityKharkovUkraine

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