Geodesic
Precise characterizations of admissible rate of decrease of a non-trivial function with zero ball means
Sbornik. Mathematics, Tome 199 (2008) no. 1, pp. 45-65 
O. A. Ochakovskaya. Precise characterizations of admissible rate of decrease of a non-trivial function with zero ball means. Sbornik. Mathematics, Tome 199 (2008) no. 1, pp. 45-65. http://geodesic.mathdoc.fr/item/SM_2008_199_1_a2/
@article{SM_2008_199_1_a2,
     author = {O. A. Ochakovskaya},
     title = {Precise characterizations of admissible rate of decrease of a~non-trivial function with zero ball means},
     journal = {Sbornik. Mathematics},
     pages = {45--65},
     year = {2008},
     volume = {199},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2008_199_1_a2/}
}
TY  - JOUR
AU  - O. A. Ochakovskaya
TI  - Precise characterizations of admissible rate of decrease of a non-trivial function with zero ball means
JO  - Sbornik. Mathematics
PY  - 2008
SP  - 45
EP  - 65
VL  - 199
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_2008_199_1_a2/
LA  - en
ID  - SM_2008_199_1_a2
ER  - 
%0 Journal Article
%A O. A. Ochakovskaya
%T Precise characterizations of admissible rate of decrease of a non-trivial function with zero ball means
%J Sbornik. Mathematics
%D 2008
%P 45-65
%V 199
%N 1
%U http://geodesic.mathdoc.fr/item/SM_2008_199_1_a2/
%G en
%F SM_2008_199_1_a2

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

Precise characterizations of an admissible rate of decrease of a non-trivial function having zero integrals over all balls of fixed radius are established. The case of an essentially anisotropic behaviour of the function at infinity is considered for the first time. In particular, the function is even allowed to grow exponentially in one variable, which is compensated in a certain sense by its rapid decrease in other variables. Bibliography: 17 titles.

[1] V. V. Volchkov, Integral geometry and convolution equations, Kluwer Acad. Publ., Dordrecht, 2003 | MR | Zbl

[2] V. V. Volchkov, “A definitive version of the local two-radii theorem”, Sb. Math., 186:6 (1995), 783–802 | DOI | MR | Zbl

[3] F. John, Plane waves and spherical means applied to partial differential equations, Intersci. Publ., New York–London, 1955 | MR | Zbl

[4] J. D. Smith, “Harmonic analysis of scalar and vector fields in $\mathbb R^n$”, Proc. Cambridge Philos. Soc., 72 (1972), 403–416 | DOI | MR | Zbl

[5] A. Sitaram, “Fourier analysis and determining sets for Radon measures on $\mathbb R^n$”, Illinois J. Math., 28:2 (1984), 339–347 | MR | Zbl

[6] S. Thangavelu, “Spherical means and CR functions on the Heisenberg group”, J. Anal. Math., 63 (1994), 255–286 | DOI | MR | Zbl

[7] V. V. Volchkov, “Solution of the support problem for several function classes”, Sb. Math., 188:9 (1997), 1279–1294 | DOI | MR | Zbl

[8] M. Shahshahani, A. Sitaram, “The Pompeiu problem in exterior domains in symmetric spaces”, Integral geometry (Brunswick, Maine, 1984), Contemp. Math., 63, Amer. Math. Soc., Providence, RI, 1987, 267–277 | MR | Zbl

[9] V. V. Volchkov, “Theorems on ball mean values in symmetric spaces”, Sb. Math., 192:9 (2001), 1275–1296 | DOI | MR | Zbl

[10] O. A. Ochakovskaya, “On functions with zero integrals over balls of fixed radius on a half-space”, Russian Acad. Sci. Dokl. Math., 64:3 (2001), 413–415 | MR | Zbl

[11] O. A. Ochakovskaya, “O funktsiyakh s nulevymi integralami po sharam fiksirovannogo radiusa”, Matem. fizika, analiz, geometriya, 9:3 (2002), 493–501 | MR | Zbl

[12] L. Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Grundlehren Math. Wiss., 256, Springer-Verlag, Berlin, 1983 | MR | MR | Zbl | Zbl

[13] G. V. Badalyan, Quasipower series and quasianalytic classes of functions, Transl. Math. Monogr., 216, Amer. Math. Soc., Providence, RI, 2002 | MR | MR | Zbl | Zbl

[14] B. G. Korenev, Vvedenie v teoriyu besselevykh funktsii, Nauka, M., 1971 | MR | Zbl

[15] E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Ser., 32, Princeton Univ. Press, Princeton, NJ, 1971 | MR | Zbl | Zbl

[16] B. Ja. Levin, Distribution of zeros of entire functions, Amer. Math. Soc., Providence, RI, 1964 | MR | MR | Zbl | Zbl

[17] A. F. Leontev, Posledovatelnosti polinomov iz eksponent, Nauka, M., 1980 | MR | Zbl