Geodesic
Asymptotically homogeneous generalized functions and some of their applications
Informatics and Automation, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 74-90

Voir la notice de l'article provenant de la source Math-Net.Ru

A brief description is given of generalized functions that are asymptotically homogeneous at the origin with respect to a multiplicative one-parameter transformation group such that the real parts of all eigenvalues of the infinitesimal matrix are positive. The generalized functions that are homogeneous with respect to such a group are described in full. Examples of the application of such functions in mathematical physics are given; in particular, they can be used to construct asymptotically homogeneous solutions of differential equations whose symbols are homogeneous polynomials with respect to such a group, as well as to study the singularities of holomorphic functions in tubular domains over cones.
Keywords: generalized functions, homogeneous functions, quasiasymptotics, partial differential equations. 
Yu. N. Drozhzhinov. Asymptotically homogeneous generalized functions and some of their applications. Informatics and Automation, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 74-90. http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a5/
@article{TRSPY_2018_301_a5,
     author = {Yu. N. Drozhzhinov},
     title = {Asymptotically homogeneous generalized functions and some of their applications},
     journal = {Informatics and Automation},
     pages = {74--90},
     year = {2018},
     volume = {301},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a5/}
}
TY  - JOUR
AU  - Yu. N. Drozhzhinov
TI  - Asymptotically homogeneous generalized functions and some of their applications
JO  - Informatics and Automation
PY  - 2018
SP  - 74
EP  - 90
VL  - 301
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a5/
LA  - ru
ID  - TRSPY_2018_301_a5
ER  - 
%0 Journal Article
%A Yu. N. Drozhzhinov
%T Asymptotically homogeneous generalized functions and some of their applications
%J Informatics and Automation
%D 2018
%P 74-90
%V 301
%U http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a5/
%G ru
%F TRSPY_2018_301_a5

[1] I. Ya. Aref'eva, I. V. Volovich, S. V. Kozyrev, “Stochastic limit method and interference in quantum many-particle systems”, Theor. Math. Phys., 183:3 (2015), 782–799 | DOI | DOI | MR | Zbl

[2] Yu. N. Drozhzhinov, “Multidimensional Tauberian theorems for generalized functions”, Russ. Math. Surv., 71:6 (2016), 1081–1134 | DOI | DOI | MR | Zbl

[3] Yu. N. Drozhzhinov, B. I. Zav'yalov, “Generalized functions asymptotically homogeneous along special transformation groups”, Sb. Math., 200:6 (2009), 803–844 | DOI | DOI | MR | Zbl

[4] Yu. N. Drozhzhinov, B. I. Zav'yalov, “Distributions asymptotically homogeneous along the trajectories determined by one-parameter groups”, Izv. Math., 76:3 (2012), 466–516 | DOI | DOI | MR | Zbl

[5] Drozhzhinov Yu. N., Zavialov B. I., “Homogeneous generalized functions with respect to one-parametric group”, $p$-Adic Numbers Ultrametric Anal. Appl., 4:1 (2012), 64–75 | DOI | MR | Zbl

[6] Yu. N. Drozhzhinov, B. I. Zavialov, “Generalized functions asymptotically homogeneous with respect to one-parametric group at origin”, Ufa Math. J., 5:1 (2013), 17–35 | DOI | MR | Zbl

[7] 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 | DOI | MR | Zbl

[8] Yu. N. Drozhzhinov, B. I. Zavialov, “Tauberian comparison theorems and hyperbolic operators with constant coefficients”, Ufa Math. J., 7:3 (2015), 47–53 | DOI | MR

[9] I. M. Gel'fand, G. E. Shilov, Generalized Functions, v. 1, Properties and Operations, Academic, New York, 1964 | MR | MR | Zbl

[10] A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439 | DOI | DOI | MR | Zbl

[11] Hörmander L., “On the division of distributions by polynomials”, Ark. Mat., 3:6 (1958), 555–568 | DOI | MR | Zbl

[12] M. O. Katanaev, “Lorentz invariant vacuum solutions in general relativity”, Proc. Steklov Inst. Math., 290 (2015), 138–142 | DOI | DOI | MR | Zbl

[13] M. O. Katanaev, “Killing vector fields and a homogeneous isotropic universe”, Phys. Usp., 59:7 (2016), 689–700 | DOI | DOI

[14] Pechen A., Trushechkin A., “Measurement-assisted Landau–Zener transitions”, Phys. Rev. A, 91:5 (2015), 052316 ; arXiv: 1506.08323[quant-ph] | DOI

[15] E. Seneta, Regularly Varying Functions, Springer, Berlin, 1976 | MR | MR | Zbl

[16] V. S. Vladimirov, Yu. N. Drozhzhinov, B. I. Zav'yalov, Tauberian Theorems for Generalized Functions, Kluwer, Dordrecht, 1988 | MR | MR | Zbl

[17] I. V. Volovich, S. V. Kozyrev, “Manipulation of states of a degenerate quantum system”, Proc. Steklov Inst. Math., 294 (2016), 241–251 | DOI | DOI | MR | Zbl

[18] V. V. Zharinov, “Bäcklund transformations”, Theor. Math. Phys., 189:3 (2016), 1681–1692 | DOI | DOI | MR | Zbl