# Complex-Valued Functions with Nondegenerate Groups of Regular Points

Complex-valued functions with nondegenerate groups of regular points are studied in the paper. A class of functions , which takes value on the complex plain and for which the limit exists, and is nonzero and finite for some from subset of positive real numbers is considered. It was received that this subset is multiplicative group, and it is called the group of regular points. Functions with nondegenerate groups of regular points generalize the class of RV functions. The corresponding limit functions are defined for complex-valued functions with nondegenerate groups of regular points. Factorization representations for this limit functions are obtained. It was shown, that for complex-valued function with nondegenerate group of regular points it limit function can be represented as a product of power function and periodic function of logarithm argument. Similar results are known for real valued functions with nondegenerate groups of regular points. Obtained results are generalized and complemented corresponding results from real valued situation. Some well-known theorems from RV functions theory can be covered from this general approach.

Publication year:
2014
Issue:
4
УДК:
517.18
P. 88–92., Refs.: 10 titles.
References:

1. J. Karamata, “Sur un mode de croissance reguliere”, Mathematica (Cluj), vol. 4, pp. 38—53, 1930.
2. N.M. Bingham et al., Regular variation. Cambridge: Cambridge University Press, 1987, 494 p.
3. Сенета Е. Правильно меняющиеся функции / Пер. с англ. — М.: Наука, 1985. — 144 с.
4. Псевдорегулярні функції та узагальнені процеси відно- влення / В.В. Булдигін, К.-Х. Індлекофер, О.І. Кле- сов, Й.Г. Штайнебах. — К.: ТВіМС, 2012. — 441 с.
5. P.D.T.A. Elliott, Probabilistic Number Theory I. New York: Springer, 1979, 393 p.
6. Заболоцький М. Комплекснозначні повільно змінні функції вздовж кривої та у вершині сектору // Мат. вісник наук. тов-ва ім. Шевченка. — 2005. — № 2. — С. 83—91.
7. V.G. Avakumovic, “Uber einer O-Inversionssatz”, Bull. Int. Acad. Youg. Sci., vol. 29-30, pp. 107—117, 1936.
8. V.V. Buldygin et al., “On factorization representation for Avakumović-Karamata functions with nondegenerate groups of regular points”, Analysis Mathematica, vol. 30, pp. 161—192, 2004.
9. Булдигін В.В., Павленков В.В. Узагальнення теореми Карамати про асимптотичну поведінку інтегралів // Теорія ймовірностей та мат. статистика. — 2009. — № 81. — С. 13—24.
10. Булдигін В.В., Павленков В.В. Теорема Караматы для регулярно LOG-периодических функций // Укр. мат. журнал. — 2012. — 64. — С. 1443—1463.

References [transliteration]:

1. J. Karamata, “Sur un mode de croissance reguliere”, Mathematica (Cluj), vol. 4, pp. 38–53, 1930.
2. N.M. Bingham et al., Regular variation. Cambridge: Cambridge University Press, 1987, 494 p.
3. Seneta E. Pravil'no meni͡ai͡ushchiesi͡a funkt͡sii / Per. s angl. – M.: Nauka, 1985. – 144 s.
4. Psevdorehuli͡arni funkt͡siï ta uzahal′neni prot͡sesy vidnovlenni͡a / V.V. Buldyhin, K.-Kh. Indlekofer, O.I. Klesov, Ĭ.H. Shtaĭnebakh. –K.: TViMS, 2012. – 441 s.
5. P.D.T.A. Elliott, Probabilistic Number Theory I. New York: Springer, 1979, 393 p.
6. Zabolot͡s′kyĭ M. Kompleksnoznachni povil′no zminni funkt͡siï vzdovz͡h kryvoï ta u vershyni sektoru // Mat. visnyk nauk. tov-va im. Shevchenka. – 2005. – # 2. – S. 83–91.
7. V.G. Avakumovic, “Uber einer O-Inversionssatz”, Bull. Int. Acad. Youg. Sci., vol. 29-30, pp. 107–117, 1936.
8. V.V. Buldygin et al., “On factorization representation for Avakumović-Karamata functions with nondegenerate groups of regular points”, Analysis Mathematica, vol. 30, pp. 161–192, 2004.
9. Buldyhin V.V., Pavlenkov V.V. Uzahal′nenni͡a teoremy Karamaty pro asymptotychnu povedinku intehraliv // Teorii͡a ĭmovirnosteĭ ta mat. statystyka. – 2009. – # 81. – S. 13–24.
10. Buldyhin V.V., Pavlenkov V.V. Teorema Karamatы dli͡a rehuli͡arno LOG-peryodycheskykh funkt͡syĭ // Ukr. mat. z͡hurnal. – 2012. – 64 . – S. 1443–1463.

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