The Regularization of Unitary Matrix Functions

Володимир Володимирович Павленков


Background. The limit behavior at infinity of unitary matrix functions of real argument and arbitrary finite dimension is considered. The question of regular variation in Karamata sense of these functions is studied.

Objective. The main aim of this work is to find the conditions under which not regularly varying unitary matrix function of real argument can be regularized by variable substitution.

Methods. It is shown in the paper that substitution \[t \to \log t \]

converts two-dimensional unitary matrix function into a power function with known matrix degree. This property is the base of all main result’s proofs.

Results. The conditions under which the unitary matrix function of arbitrary finite dimension can be regularized are obtained.

Conclusions. A significant difference between the matrix functions of dimension lower than 4 and the matrix functions of higher dimension is established. The constructed example of unitary matrix function of 4-dimension that can’t be regularized by substitution \[t \to \log t \] shows this difference.


Regularly varying function; Matrix function; Linear operator


J. Karamata, “On a method of regularly growth”, Mathematica (Cluj), no. 4, pp. 3853, 1930.

N.M. Bingham et al., Regular Variation. Cambridge, UK: Cambridge University Press, 1987.

E. Seneta, Regularly Varying Functions. Moscow, USSR, Nauka, 1985 (in Russian).

V.V. Buldygin et al., Pseudo Regular Function and Generalized Renewal Processes. Kyiv, Ukraine: TViMS, 2012 (in Ukrainian).

V.V. Buldygin and V.V. Pavlenkov, “On a generalization of Karamata’s theorem on the asymptotic behavior of integrals”, Teoriya Imovirnostej i Matematychna Statystyka, no. 81, pp. 13–24, 2009 (in Ukrainian).

V.V. Buldygin and V.V. Pavlenkov, “Karamata’s theorem for regularly LOG-periodic functions”, Ukr. Matematychnyj Zhurnal, no. 64, pp. 1443–1463, 2012 (in Russian).

V.V. Pavlenkov, “Complex-valued functions with nondegenerate group of regular points”, Naukovi Visti NTUU KPI, no. 4, pp. 88–92, 2014 (in Ukrainian).

M.M. Meerschaert and H.-P. Scheffler, Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice (Willey Series in Probability and Statistics). John Wiley & Sons, 2001.

F.R. Gantmaher, Matrixes Theory. Moscow, USSR: Nauka, 1967 (in Russian).

GOST Style Citations

  1. Karamata J. Sur un mode de croissance reguliere // Mathematica (Cluj). – 1930.  4.  P. 3853.

  2. Bingham N.M., Goldie C.M., Teugels J.L. 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 відновлення / В.В. Булдигін, К.-Х. Індлекофер, О.І. Клесов, Й.Г. Штайне­бах. – К.: ТВіМС, 2012.  441 с.

  5. Булдигін В.В., Павленков В.В. Узагальнення теореми Карамати про асимптотичну поведінку інтегралів // Теорія ймовірностей та мат. статистика.  2009.  № 81.  С. 1324.

  6. Булдигин В.В., Павленков В.В. Теорема Караматы для регулярно LOG-периодических функций // Укр. мат. журнал.  2012.  64.  С. 14431463.

  7. Павленков В.В. Комплекснозначні функції з невиродженими групами регулярних точок // Наукові вісті НТУУ “КПІ”.  2014.  № 4.  С. 88 92.

  8. Meerschaert M.M., Scheffler H.-P. Limit distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. – John Wiley & Sons, 2001. – 512 p. – (Willey Series in Probability and Statistics).

  9. Гантмахер Ф.Р. Теория матриц. – М.: Наука, 1967. – 576 с.

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