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 Ufimsk. Mat. Zh., 2009, Volume 1, Issue 4, Pages 24–57 (Mi ufa30)

Asymptotically quasi-homogeneous generalized functions at the origine

Yu. N. Drozhzhinov, B. I. Zavialov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Generalized functions having quasiasymptotics along special groups of transformations of independent variables in the asymptotic scale of regularly varying functions are said to be asymptotically homogeneous along these transformations groups. In particular, all “quasihomogeneous” distributions have this property. A complete description of asymptotically homogeneous in the origin distributions along a transformation group determined by a vector $a\in\mathbb R_+^n$ is obtained (including the case of critical orders). Special distribution spaces are introduced and investigated to this end. The results obtained in the paper are applied for construction of asymptotically quasihomogeneous solutions of differential equations whose symbols are quasihomogeneous polynomials.

Keywords: Generalized Functions, Asymptotically homogeneous functions, tauberian theorems, quasiasymptotic form, regylary varying functions.

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Bibliographic databases:
UDC: 517.5

Citation: Yu. N. Drozhzhinov, B. I. Zavialov, “Asymptotically quasi-homogeneous generalized functions at the origine”, Ufimsk. Mat. Zh., 1:4 (2009), 24–57

Citation in format AMSBIB
\Bibitem{DroZav09} \by Yu.~N.~Drozhzhinov, B.~I.~Zavialov \paper Asymptotically quasi-homogeneous generalized functions at the origine \jour Ufimsk. Mat. Zh. \yr 2009 \vol 1 \issue 4 \pages 24--57 \mathnet{http://mi.mathnet.ru/ufa30} \zmath{https://zbmath.org/?q=an:05972408} 

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This publication is cited in the following articles:
1. Drozhzhinov Yu.N. Zav'yalov B.I., “Asymptotically Homogeneous Solutions of Differential Equations Whose Symbols Are Polynomials Quasi-Homogeneous with Respect to One-Parameter Groups with Generators Containing a Nilpotent Component”, Dokl. Math., 88:2 (2013), 590–592
2. Yu. N. Drozhzhinov, B. I. Zavialov, “Asymptotically homogeneous solutions to differential equations with homogeneous polynomial symbols with respect to a multiplicative one-parameter group”, Proc. Steklov Inst. Math., 285 (2014), 99–119
3. Baizaev S., Rakhimova M.A., “Some Functional Equations in Schwartz Space and Their Applications”, Ufa Math. J., 10:1 (2018), 3–13
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