Associated and quasi associated homogeneous distributions (generalized functions)
In this paper analysis of the concept of associated homogeneous distributions (generalized functions) is given, and some problems related to these distributions are solved. It is proved that (in the one-dimensional case) there exist only associated homogeneous distributions of order k = 1 . We intro...
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| Vydáno v: | Journal of mathematical analysis and applications Ročník 338; číslo 1; s. 48 - 70 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier Inc
01.02.2008
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| Témata: | |
| ISSN: | 0022-247X, 1096-0813 |
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| Abstract | In this paper analysis of the concept of
associated homogeneous distributions (generalized functions) is given, and some problems related to these distributions are solved. It is proved that (in the one-dimensional case) there exist
only associated homogeneous distributions of order
k
=
1
. We introduce a definition of
quasi associated homogeneous distributions which is a natural generalization of the notion of
associated homogeneous distributions and provide a mathematical description of all quasi associated homogeneous distributions and their Fourier transforms. It is proved that the class of
quasi associated homogeneous distributions coincides with the class of distributions introduced in [I.M. Gel'fand, G.E. Shilov, Generalized Functions, vol. 1, Properties and Operations, Academic Press, New York, 1964, Chapter I, §4] as the class of
associated homogeneous distributions. For the multidimensional case it is proved that
f is a
quasi associated homogeneous distribution if and only if it satisfies an Euler type system of differential equations. A new type of Γ-functions generated by quasi associated homogeneous distributions is defined. |
|---|---|
| AbstractList | In this paper analysis of the concept of
associated homogeneous distributions (generalized functions) is given, and some problems related to these distributions are solved. It is proved that (in the one-dimensional case) there exist
only associated homogeneous distributions of order
k
=
1
. We introduce a definition of
quasi associated homogeneous distributions which is a natural generalization of the notion of
associated homogeneous distributions and provide a mathematical description of all quasi associated homogeneous distributions and their Fourier transforms. It is proved that the class of
quasi associated homogeneous distributions coincides with the class of distributions introduced in [I.M. Gel'fand, G.E. Shilov, Generalized Functions, vol. 1, Properties and Operations, Academic Press, New York, 1964, Chapter I, §4] as the class of
associated homogeneous distributions. For the multidimensional case it is proved that
f is a
quasi associated homogeneous distribution if and only if it satisfies an Euler type system of differential equations. A new type of Γ-functions generated by quasi associated homogeneous distributions is defined. |
| Author | Shelkovich, V.M. |
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| Cites_doi | 10.1016/j.jmaa.2005.05.016 10.1023/A:1006504318696 |
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| Keywords | Associated homogeneous functions Euler type system of differential equations Quasi associated homogeneous functions Distributions |
| Language | English |
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| References | Estrada, Kanwal (bib005) 2002 Albeverio, Khrennikov, Shelkovich (bib002) 2003; 393 Vladimirov, Drozzinov, Zavyalov (bib011) 1988 Drozzinov, Zavialov (bib003) 1977; 102 von Grudzinski (bib007) 1991 Gel'fand, Shilov (bib006) 1964 Hörmander (bib008) 1983 Albeverio, Khrennikov, Shelkovich (bib001) 2006; 313 Estrada, Kanwal (bib004) 1994 Lesch (bib009) 1999; 17 Vilenkin (bib010) 1972 Gel'fand (10.1016/j.jmaa.2007.04.069_bib006) 1964 Albeverio (10.1016/j.jmaa.2007.04.069_bib001) 2006; 313 Vladimirov (10.1016/j.jmaa.2007.04.069_bib011) 1988 Albeverio (10.1016/j.jmaa.2007.04.069_bib002) 2003; 393 Lesch (10.1016/j.jmaa.2007.04.069_bib009) 1999; 17 Vilenkin (10.1016/j.jmaa.2007.04.069_bib010) 1972 Estrada (10.1016/j.jmaa.2007.04.069_bib004) 1994 von Grudzinski (10.1016/j.jmaa.2007.04.069_bib007) 1991 Drozzinov (10.1016/j.jmaa.2007.04.069_bib003) 1977; 102 Hörmander (10.1016/j.jmaa.2007.04.069_bib008) 1983 Estrada (10.1016/j.jmaa.2007.04.069_bib005) 2002 |
| References_xml | – year: 2002 ident: bib005 article-title: A Distributional Approach to Asymptotics: Theory and Applications – volume: 393 start-page: 300 year: 2003 end-page: 303 ident: bib002 article-title: Associated homogeneous publication-title: Dokl. Akad. Nauk – year: 1964 ident: bib006 article-title: Generalized Functions, vol. 1: Properties and Operations – year: 1991 ident: bib007 article-title: Quasihomogeneous Distributions – volume: 102 start-page: 372 year: 1977 end-page: 390 ident: bib003 article-title: Quasi-asymptotics of generalized functions and Tauberian theorems in the complex domain publication-title: Math. Sb. – year: 1994 ident: bib004 article-title: Asymptotic Analysis: A Distributional Approach – year: 1972 ident: bib010 article-title: Generalized functions publication-title: Funktsional. Anal. – year: 1988 ident: bib011 article-title: Tauberian Theorems for Generalized Functions – year: 1983 ident: bib008 article-title: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis – volume: 17 start-page: 151 year: 1999 end-page: 187 ident: bib009 article-title: On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols publication-title: Ann. Global Anal. Geom. – volume: 313 start-page: 64 year: 2006 end-page: 83 ident: bib001 article-title: Associated homogeneous publication-title: J. Math. Anal. Appl. – year: 1972 ident: 10.1016/j.jmaa.2007.04.069_bib010 article-title: Generalized functions – volume: 313 start-page: 64 year: 2006 ident: 10.1016/j.jmaa.2007.04.069_bib001 article-title: Associated homogeneous p-adic distributions publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2005.05.016 – year: 1964 ident: 10.1016/j.jmaa.2007.04.069_bib006 – volume: 17 start-page: 151 issue: 2 year: 1999 ident: 10.1016/j.jmaa.2007.04.069_bib009 article-title: On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols publication-title: Ann. Global Anal. Geom. doi: 10.1023/A:1006504318696 – year: 1983 ident: 10.1016/j.jmaa.2007.04.069_bib008 – volume: 102 start-page: 372 issue: 3 year: 1977 ident: 10.1016/j.jmaa.2007.04.069_bib003 article-title: Quasi-asymptotics of generalized functions and Tauberian theorems in the complex domain publication-title: Math. Sb. – year: 1991 ident: 10.1016/j.jmaa.2007.04.069_bib007 – volume: 393 start-page: 300 issue: 3 year: 2003 ident: 10.1016/j.jmaa.2007.04.069_bib002 article-title: Associated homogeneous p-adic generalized functions publication-title: Dokl. Akad. Nauk – year: 1994 ident: 10.1016/j.jmaa.2007.04.069_bib004 – year: 2002 ident: 10.1016/j.jmaa.2007.04.069_bib005 – year: 1988 ident: 10.1016/j.jmaa.2007.04.069_bib011 |
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associated homogeneous distributions (generalized functions) is given, and some problems related to these... |
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| SubjectTerms | Associated homogeneous functions Distributions Euler type system of differential equations Quasi associated homogeneous functions |
| Title | Associated and quasi associated homogeneous distributions (generalized functions) |
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