Spaces of analytic functions of prescribed growth near the boundary

V. V. Napalkov

Izvestiya. Mathematics , Tome 30 (1988) no. 2, pp. 263-281.

Résumé

Let $D$ be an arbitrary bounded convex domain in the plane $\mathbf C$. For a certain sequence of convex functions $\varphi=\{\varphi_j\}_{j=1}^\infty$, $\varphi_j(z)\geqslant\varphi_{j+1}(z)$, given on $D$ the space $H_\varphi (D)$ is constructed as the projective limit of the normed spaces $$ H_j(D)=\{f(z)\in H(D):\|f\|_j=\sup_D|f(z)|\exp{(-\varphi_j(z))}\infty\},\qquad j=1,2,\dots, $$ where $H(D)$ is the space of analytic functions on $D$. The space $H_\varphi^*(D)$ is described in terms of Laplace transforms. A special role in this description is played by a generalization, proved in the article, of the Paley–Wiener theorem to the case of spaces of infinitely differentiable functions with prescribed growth near the boundary. The result is used in questions involving expansions of functions in Dirichlet series. Figures: 1. Bibliography: 17 titles.

Bibliographie

[1] Leontev A. F., “Ryady eksponent dlya funktsii s opredelennym rostom vblizi granitsy”, Izv. AN SSSR. Ser. matem., 44:6 (1980), 1308–1328 | MR

[2] Khërmander L., Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, Mir, M., 1968 | MR

[3] Yulmukhametov R. S., “Prostranstva analiticheskikh funktsii, imeyuschikh zadannyi rost vblizi granitsy”, Matem. zametki, 32:1 (1982), 41–57 | MR | Zbl

[4] Derzhavets B. A., “Prostranstva funktsii, analiticheskikh v vypuklykh oblastyakh $\mathbf{C}^n$ i imeyuschikh zadannoe povedenie vblizi granitsy”, Dokl. AN SSSR, 276:6 (1984), 1297–1300 | MR

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

[6] Yulmukhametov R. S., “Dostatochnye mnozhestva v odnom prostranstve tselykh funktsii”, Matem. sb., 116(158):3 (1981), 427–439 | MR | Zbl

[7] Napalkov V. V., “O diskretnykh dostatochnykh mnozhestvakh v nekotorykh prostranstvakh tselykh funktsii”, Dokl. AN SSSR, 250:4 (1980), 809–812 | MR | Zbl

[8] Yulmukhametov R. S., “Priblizhenie subgarmonicheskikh funktsii”, Matem. sb., 124(166):3 (1984), 393–415 | MR | Zbl

[9] Yulmukhametov R. S., “Approksimatsiya subgarmonicheskikh funktsii”, Analysis Mathematica, 11 (1985), 257–282 | DOI | MR | Zbl

[10] Sebashtyan-i-Silva Zh., “O nekotorykh klassakh lokalno vypuklykh prostranstv, vazhnykh v prilozheniyakh”, Matematika, I:1 (1957), 60–77

[11] Napalkov V. V., Uravneniya svertki v mnogomernykh prostranstvakh, Nauka, M., 1982 | MR

[12] Malgranzh B., Idealy differentsialnykh funktsii, Mir, M., 1968

[13] Rokafellar R., Vypuklyi analiz, Mir, M., 1973

[14] Khërmander L., Lineinye differentsialnye operatory s chastnymi proizvodnymi, Mir, M., 1965 | MR

[15] Vladimirov V. S., Metody teorii funktsii mnogikh kompleksnykh peremennykh, Nauka, M., 1964 | MR

[16] Ehrenpreis L., Fourier analysis several complex variables, Wiley–Interscience publishers, New York, 1970 | Zbl

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