SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2008 / Cilt: 37 - Sayı: 1

On the composition of the distributions $x^{-1}ln^m|x|$ and $x^r$

Brian FISHER
İnci EGE
Emin ÖZÇAĞ
Pages
1–8
Publication Date
2022-10-18
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

The composition of the distributions $x^{-1} ln^m |x|$ and $x^r$ is evaluated for r,m = 1, 2, 3 . . ..

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Let F be a distribution and f a locally summable function.
📊
Key Findings & Data:The dis- tribution F(f) is defined as the neutrix limit of the sequence ${F_n(f)}$, where $F_n(x) = F(x)ast delta_n(x)$ and ${delta_n(x)}$ is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function $delta(x)$.
💡
Conclusion & Impact:The composition of the distributions $x^{-1} ln^m |x|$ and $x^r$ is evaluated for r,m = 1, 2, 3 . . ..
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

Let F be a distribution and f a locally summable function. The dis- tribution F(f) is defined as the neutrix limit of the sequence ${F_n(f)}$, where $F_n(x) = F(x)ast delta_n(x)$ and ${delta_n(x)}$ is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function $delta(x)$. The composition of the distributions $x^{-1} ln^m |x|$ and $x^r$ is evaluated for r,m = 1, 2, 3 . . ..

🔖

Cite This Article

FISHER, B., EGE, İ., ÖZÇAĞ, E. (2022). On the composition of the distributions $x^{-1}ln^m|x|$ and $x^r$. Hacettepe Journal of Mathematics and Statistics, Cilt: 37 - Sayı: 1, 1–8. https://ajindex.com/articles/530719

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (10)

  1. van der Corput, J.G. Introduction to the neutrix calculus, J. Analyse Math. 7, 291–398, 1959
  2. Fisher, B. On defining the distribution $delta^{(r)}(f(x))$, Rostock. Math. Kolloq. 23, 73–80, 1993
  3. Fisher, B. On defining the change of variable in distributions, Rostock. Math. Kolloq. 28, 75–86, 1985
  4. Fisher, B., Jolevska-Tuneska, B. and Takaci, A. On the composition of the distributions $x^{-s}$ In |x| and $x^r$, Thai J. Math. 1, 39–48, 2003
  5. Gel’fand, I.M. and Shilov, G. Generalized Functions, Vol. I (Academic Press, 1964)
  6. Koh, E. L. and Li, C.K. On Distributions $delta^k$ and $(delta^{'})^k$, Math. Nachr. 157, 243–248, 1992
  7. Kou, H. and Fisher, B. On Composition of Distributions, Publ. Math. Debrecen 40 (3-4), 279–290, 1992
  8. Nicholas, J.D. and Fisher, B. A result on the composition of distributions, Proc. Indian Acad. Sci. Math. Sci. 109 (3), 317–323, 1999
  9. Özçağ, E. Defining the kth powers of the Dirac Delta distribution for negative integers, Appl. Math. Letters 14, 419–423, 2001
  10. Özçağ, E., Ege, I and Gürçay, H. On powers of the Heaviside Function for negative integers, J. Math. Anal. Appl. 326, 101–107, 2007.