On sufficient sets in spaces of entire functions of several variables

A. B. Sekerin

Sbornik. Mathematics, Tome 64 (1989) no. 1, pp. 263-276

Résumé

The main result is Theorem 1. {\it Let $D$ be a bounded convex domain in $\mathbf C^n,$ $n\geqslant2,$ with $0\in D$. Let $H(z)=\max_{\lambda\in\overline D}\mathbf{Re}\langle\lambda,z\rangle$. Let $L(z)$ be an entire function of exponential type whose zero set $S$ is the union of planes $P_m=\{z:\langle a_m,z\rangle=c_m\},$ $m\in\mathbf N,$ $|a_m|=1$. Suppose the following conditions hold}: a) {\it there exist constants $c,$ $r_0,$ $d_0,$ $\gamma\in(0,1),$ such that the estimate $$ \left|\ln|L(z)|-H(z)\right|\leqslant c\left|\ln d\right||z|^{1-\gamma} $$ holds if the point $z\in\mathbf C^n,$ satisfies $|z|\geqslant r_0,$ $\inf_{w\in S}|z-w|=d(z,S)\geqslant d>0,$ $d<d_0$}; b) {\it for every $m$ the restriction of the entire function $(\langle a_m,z\rangle-c_m)^{-1}L(z)$ to the plane $P_m$ is not identically zero}; c) {\it there exist constants $c$ and $N$ such that for $m\ne k$ either $d(P_m,P_k)\geqslant c|c_m|^{-N}|c_k|^{-N}$ or $1-|\langle a_m,\overline a_k\rangle|\geqslant c|c_m|^{-N}|c_k|^{-N}$. Then every analytic function $f(z)$ in the domain $D$ can be represented by a series $$ f(z)=\sum_{m=1}^\infty\int_{P_m}\exp\langle\lambda,z\rangle\,d\mu_m(\lambda) $$ converging in the topology of $H(D)$.} Bibliography: 11 titles.

Bibliographie

[1] Leontev A. F., Ryady eksponent, Nauka, M., 1976 | MR

[2] Korobeinik Yu. F., “Predstavlyayuschie sistemy”, Izv. AN SSSR. Ser. matem., 42 (1978), 325–355 | MR | Zbl

[3] Leontev A. F., “O predstavlenii analiticheskikh funktsii ryadami eksponent v politsilindricheskoi oblasti”, Matem. sb., 89(131) (1972), 586–598 | MR

[4] Schneider D. M., “Sufficient sets for some spaces of entire functions”, Trans. Amer. Math. Soc., 197 (1974), 161–180 | DOI | MR | Zbl

[5] Morzhakov V. V., Absolyutno predstavlyayuschie sistemy eksponent v prostranstvakh analiticheskikh funktsii mnogikh kompleksnykh peremennykh, Rukop. dep. v VINITI 16 yanv. 1981. No 245–81 Dep

[6] Napalkov V. V., Sekerin A. B., “Slabo dostatochnye mnozhestva i predstavlenie analiticheskikh funktsii mnogikh kompleksnykh peremennykh ryadami Dirikhle”, DAN SSSR, 260:3 (1981), 535–539 | MR | Zbl

[7] Napalkov V. V., “O sravnenii topologii v nekotorykh prostranstvakh tselykh funktsii”, DAN SSSR, 264:4 (1982), 827–830 | MR | Zbl

[8] Yulmukhametov R. S., “Priblizhenie subgarmonicheskikh funktsii”, Matem. sb., 124(166) (1984), 393–415 | MR | Zbl

[9] Napalkov V. V., “O diskretnykh slabo dostatochnykh mnozhestvakh v nekotorykh prostranstvakh tselykh funktsii”, Izv. AN SSSR. Ser. matem., 45 (1981), 1088–1099 | MR | Zbl

[10] Levin B. Ya., Raspredelenie kornei tselykh funktsii, Gostekhizdat, M., 1956

[11] Chirka E. M., Kompleksnye analiticheskie mnozhestva, Nauka, M., 1985 | MR