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2001 | 21 | 2 | 149-157
Tytuł artykułu

Simple fractions and linear decomposition of some convolutions of measures

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EN
Abstrakty
EN
Every characteristic function φ can be written in the following way:
φ(ξ) = 1/(h(ξ) + 1), where h(ξ) =
⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0

⎩ ∞ if φ(ξ) = 0
This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form $φ_{a}(ξ) = [a/(h(ξ) + a)]$, where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain that convolutions of measures $μ_{a}$ with $μ̂_{a}(ξ) = φ_{a}(ξ)$ are linear combinations of powers of such measures. This can simplify calculations. It is interesting that this simplification uses signed measures since coefficients of linear combinations can be negative numbers. All the results of this paper except Proposition 1 remain true if we replace probability measures with complex valued measures with finite variation, and replace the characteristic function with Fourier transform.
Twórcy
  • Institute of Mathematics, University of Zielona Góra, Podgórna 50, 65-246 Zielona Góra, Poland
autor
  • Facultteit Informatietechnologie en System, Technische Universiteit Delft, Mekelweg 4, Postbus 5031, 2600 GA Delft, Holland
Bibliografia
  • [1] W. Feller, An Introduction to Probability Theory and its Application, volume II. Wiley, New York 1966.
  • [2] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Fifth Edition, Academic Press 1997.
  • [3] N. Jacobson, Basic Algebra I, W.H. Freeman and Company, San Francisco 1974.
  • [4] S. Lang, Algebra, Addison-Weslay 1970, Reading USA, Second Edition.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1026
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