The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros is extended to subharmonic functions in , having the Riesz masses on a ray.
Le théorème de Valiron-Titchmarsh sur le comportement asymptotique des fonctions entières avec des zéros négatifs est étendu aux fonctions sous-harmoniques dans , ayant les masses de Riesz sur un rayon.
@article{AFST_2014_6_23_1_159_0,
author = {Alexander I. Kheyfits},
title = {Valiron-Titchmarsh {Theorem} for {Subharmonic} {Functions} in ${\mathbb{R}}^n$ {With} {Masses} on a {Half-Line}},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {159--173},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 23},
number = {1},
year = {2014},
doi = {10.5802/afst.1401},
zbl = {1295.31014},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1401/}
}
TY - JOUR
AU - Alexander I. Kheyfits
TI - Valiron-Titchmarsh Theorem for Subharmonic Functions in ${\mathbb{R}}^n$ With Masses on a Half-Line
JO - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY - 2014
SP - 159
EP - 173
VL - 23
IS - 1
PB - Université Paul Sabatier, Institut de Mathématiques
PP - Toulouse
UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1401/
DO - 10.5802/afst.1401
LA - en
ID - AFST_2014_6_23_1_159_0
ER -
%0 Journal Article
%A Alexander I. Kheyfits
%T Valiron-Titchmarsh Theorem for Subharmonic Functions in ${\mathbb{R}}^n$ With Masses on a Half-Line
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2014
%P 159-173
%V 23
%N 1
%I Université Paul Sabatier, Institut de Mathématiques
%C Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1401/
%R 10.5802/afst.1401
%G en
%F AFST_2014_6_23_1_159_0
Alexander I. Kheyfits. Valiron-Titchmarsh Theorem for Subharmonic Functions in ${\mathbb{R}}^n$ With Masses on a Half-Line. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 1, pp. 159-173. doi : 10.5802/afst.1401. https://afst.centre-mersenne.org/articles/10.5802/afst.1401/
[1] Agranovich (P.Z.).— Polynomial asymptotic representations of subharmonic functions with masses on one ray in the space (Ukrainian). Matematychni Studii, Lviv. 23, p. 169-178 (2005). | MR | Zbl
[2] Agranovich (P.Z.), Logvinenko (V.N.).— An analog of the Valiron-Titchmarsh theorem for two-term asymptotics of subharmonic functions with masses on a finite system of rays. Sib. Math. J. 5, p. 3-19 (1985). | MR | Zbl
[3] Azarin (V.S.).— Indicator of a function subharmonic in n-dimensional space (Russian). Dokl. Akad. Nauk SSSR, 139, p. 1033-1036 (1961). | MR | Zbl
[4] Azarin (V.S.).— On the Valiron-Titchmarsh theorem and limit sets of entire functions. Proceedings of the Ashkelon Workshop on Complex Function Theory (1996), 5360. Israel Math. Conf. Proc., 11, Bar-Ilan Univ., Ramat Gan (1997). | MR | Zbl
[5] Azarin (V.S.).— Growth Theory of Subharmonic Functions. Birkh¨auser, Basel – Boston – Berlin (2009). | MR | Zbl
[6] Bateman (H.), Erdélyi.— Higher Transcendental Functions, Vol. 1. McGraw-Hill, New York – Toronto – London (1953). | MR | Zbl
[7] Bateman (H.), Erdélyi.— Tables of Integral Transforms, Vol. 1. McGraw-Hill, New York – Toronto – London (1954). | Zbl
[8] Bauer (H.F.).— Tables of the roots of the associated Legendre function with respect to the Degree, Mathematics of Computation, 46, No. 174, p. 601-602 (1986). | MR | Zbl
[9] Beardon (A.F.).— Montel’s theorem for subharmonic functions and solutions of partial differential equations. Proc. Camb. Phil. Soc. 69, p. 123-150 (1971). | MR | Zbl
[10] Delange (H.).— Un théorème sur les fonctions entières à zéros réelles et négative. J. Math. Pures Appl. (9) 31, p. 55-78 (1952). | Zbl
[11] Drasin (D.).— Baernstein’s thesis and entire functions with negative zeros. Matematychni Studii, 34, p. 160-167 (2010). | MR | Zbl
[12] Goldberg (A.A.), Ostrovskii (I.V.).— On the growth of a subharmonic function with Riesz measure on a ray. Matematicheskaya Fizika, Analiz, Geometriya, 11, p. 107-113 (2004). | MR | Zbl
[13] Hardy (G.H.).— Divergent Series. Oxford (1949). | MR | Zbl
[14] Hayman (W.K.), Kennedy (P.B.).— Subharmonic Functions. Vol. 1. Academic Press, London – New York – San Francisco (1976). | MR | Zbl
[15] Hobson (E.W.).— The Theory of Spherical and Ellipsoidal Harmonics. Chelsey, New York (1955). | MR
[16] Kheyfits (A.).— A generalization of E. Titchmarsh theorem on entire functions with negative zeros. Izv. VUZov. Math. No. 2 (129) p. 99-105 (1973). | MR
[17] Kheyfits (A.).— Analogue of the Valiron-Titchmarsh theorem for entire functions with roots on a logarithmic spiral, Soviet Math. (Izv. VUZ.) 24, p. 92-94 (1980). | Zbl
[18] Levin (B.Ya.).— Distribution of Zeros of Entire Functions. Translation of Mathematical Monographs, Vol. 5. Amer. Math. Soc., Providence, Rhode Island, Revised Edition (1980). | MR | Zbl
[19] Levin (B.Ya.).— Lectures on Entire Functions. Amer. Math. Soc., Providence, Rhode Island (1996). | MR | Zbl
[20] Paley (R.E.A.C.), Wiener (N.).— Fourier Transforms in the Complex Domain, Amer. Math. Soc., New York (1934). | MR | Zbl
[21] Ronkin (L.I.).— Functions of Completely Regular Growth. Kluwer Acad. Publ., Dordrecht (1992). | MR | Zbl
[22] Szmytkowski (R.).— Some integrals and series involving the Gegenbauer polynomials and the Legendre functions on the cut (-1, 1). Integral Transforms and Special Functions, 23, p. 847-852 (2012). | MR | Zbl
[23] Titchmarsh (E.C.).— On integral functions with real negative zeros. Proc. London Math. Soc. 26, p. 185-200 (1927). | MR
[24] Valiron (G.).— Sur les fonctions entières d’ordre nul et d’ordre fini et en particulier les fonctions à correspondance régulière. Annales de la faculté des sciences de Toulouse (3) 5, p. 117-257 (1913). | MR
Cited by Sources: