Some results on the product of distributions and the change of variable

Emin Özçag; Brian Fisher

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 4, page 677-685
  • ISSN: 0010-2628

Abstract

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Let F and G be distributions in 𝒟 ' and let f be an infinitely differentiable function with f ' ( x ) > 0 , (or < 0 ). It is proved that if the neutrix product F ∘ G exists and equals H , then the neutrix product F ( f ) ∘ G ( f ) exists and equals H ( f ) .

How to cite

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Özçag, Emin, and Fisher, Brian. "Some results on the product of distributions and the change of variable." Commentationes Mathematicae Universitatis Carolinae 32.4 (1991): 677-685. <http://eudml.org/doc/247323>.

@article{Özçag1991,
abstract = {Let $F$ and $G$ be distributions in $\mathcal \{D\}^\{\prime \}$ and let $f$ be an infinitely differentiable function with $f^\{\prime \}(x)>0$, (or $<0$). It is proved that if the neutrix product $F\circ G$ exists and equals $H$, then the neutrix product $F(f)\circ G(f)$ exists and equals $H(f)$.},
author = {Özçag, Emin, Fisher, Brian},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {distribution; neutrix product; change of variable; change of variable; Dirac delta distribution; neutrix product},
language = {eng},
number = {4},
pages = {677-685},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Some results on the product of distributions and the change of variable},
url = {http://eudml.org/doc/247323},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Özçag, Emin
AU - Fisher, Brian
TI - Some results on the product of distributions and the change of variable
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 4
SP - 677
EP - 685
AB - Let $F$ and $G$ be distributions in $\mathcal {D}^{\prime }$ and let $f$ be an infinitely differentiable function with $f^{\prime }(x)>0$, (or $<0$). It is proved that if the neutrix product $F\circ G$ exists and equals $H$, then the neutrix product $F(f)\circ G(f)$ exists and equals $H(f)$.
LA - eng
KW - distribution; neutrix product; change of variable; change of variable; Dirac delta distribution; neutrix product
UR - http://eudml.org/doc/247323
ER -

References

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  1. van der Corput J.G., Introduction to the neutrix calculus, J. Analyse Math. 7 (1959-60), 291-398. (1959-60) Zbl0097.10503MR0124678
  2. Fisher B., A non-commutative neutrix product of distributions, Math. Nachr. 108 (1982), 117-127. (1982) Zbl0522.46025
  3. Fisher B., On defining the distribution δ ( r ) ( f ( x ) ) for summable f , Publ. Math. Debrecen 32 (1985), 233-241. (1985) MR0834774
  4. Fisher B., On the product of distributions and the change of variable, Publ. Math. Debrecen 35 (1988), 37-42. (1988) Zbl0668.46015MR0971950
  5. Fisher B., Özcağ E., A result on distributions and the change of variable, submitted for publication. 
  6. Gel'fand I.M., Shilov G.E., Generalized Functions, vol. I., Academic Press, 1964. Zbl0159.18301MR0166596

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