Geodesic
Universal Symbols on Locally Compact Abelian Groups
Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 1, pp. 1-16.

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

From the viewpoint of elementary functional analysis, Bernstein inequalities are mainly sharp estimates for the norms of certain operators of convolution of entire functions bounded on the real line and having finite exponential type not exceeding a given one with (complex) Borel measures of finite total variation. If we assume that the functions are defined on a locally compact Abelian group and use the uniform norms, then the generalized Bernstein spaces are parametrized by compact sets in the dual group $X$ and the symbols of the operators are the restrictions to compact sets in $X$ of functions locally coinciding with the Fourier transforms of measures. There exists symbols such that, in the case of uniform norms (and then, as it turns out, also in more general cases), the norm of the corresponding operator coincides with its spectral radius. The main result of the paper is a description of these (universal) symbols in terms of positive definite functions. Connected groups play a special role here.
Keywords: complex Banach algebra, locally compact Abelian group, Bernstein inequalities, positive definite function, spectrum. 
@article{FAA_2013_47_1_a0,
     author = {E. A. Gorin and S. Norvidas},
     title = {Universal {Symbols} on {Locally} {Compact} {Abelian} {Groups}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {1--16},
     publisher = {mathdoc},
     volume = {47},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2013_47_1_a0/}
}
TY  - JOUR
AU  - E. A. Gorin
AU  - S. Norvidas
TI  - Universal Symbols on Locally Compact Abelian Groups
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2013
SP  - 1
EP  - 16
VL  - 47
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2013_47_1_a0/
LA  - ru
ID  - FAA_2013_47_1_a0
ER  - 
%0 Journal Article
%A E. A. Gorin
%A S. Norvidas
%T Universal Symbols on Locally Compact Abelian Groups
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2013
%P 1-16
%V 47
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2013_47_1_a0/
%G ru
%F FAA_2013_47_1_a0
E. A. Gorin; S. Norvidas. Universal Symbols on Locally Compact Abelian Groups. Funkcionalʹnyj analiz i ego priloženiâ, Tome 47 (2013) no. 1, pp. 1-16. http://geodesic.mathdoc.fr/item/FAA_2013_47_1_a0/

[1] B. Ya. Levin, Lectures on Entier Functions. In collaboration with Yu. Lyubarskii, M. Sodin, and V. Tkachenko, Transl. Math. Monographs, 150, Amer. Math. Soc., Providence, RI, 1996 | DOI | MR | Zbl

[2] N. Burbaki, Spektralnaya teoriya, Mir, M., 1972 | MR

[3] E. A. Gorin, “Ob issledovaniyakh G. E. Shilova po teorii kommutativnykh banakhovykh algebr i ikh dalneishem razvitii”, UMN, 33:4 (202) (1978), 169–188 | MR | Zbl

[4] S. T. Norvidas, “Ob ustoichivosti differentsialnykh operatorov v prostranstvakh tselykh funktsii”, Dokl. AN SSSR, 291:3 (1986), 548–551 | MR | Zbl

[5] E. A. Gorin, “Universal symbols on locally compact Abelian groups”, Bull. Polish Acad. Sci. Math., 51:2 (2003), 199–204 | MR | Zbl

[6] W. Rudin, Fourier Analysis on Groups, Interscience, New York, 1967 | MR

[7] Yu. I. Lyubich, V. I. Matsaev, G. M. Feldman, “O predstavleniyakh s otdelimym spektrom”, Funkts. analiz i ego pril., 7:2 (1973), 52–61 | MR | Zbl

[8] E. Khyuitt, K. Ross, Abstraktnyi garmonicheskii analiz, v. 1, Nauka, M., 1975

[9] E. A. Gorin, “Funktsionalno-algebraicheskii variant teoremy Bora–van Kampena”, Matem. sb., 82(124):2(6) (1970), 260–272 | MR