Balayage on a system of rays and entire functions of completely regular growth

B. N. Khabibullin

Izvestiya. Mathematics , Tome 38 (1992) no. 1, pp. 179-197.

Résumé

This paper presents a technique for constructing functions that are subharmonic in the complex plane, agree with a given subharmonic function $u$ on a system $S$ of rays with vertex at the origin, and are harmonic outside $S$. For a wide class of systems $S$, this technique permits one to obtain criteria for the complete regularity of growth of entire functions $f$ on $S$ in terms of the balayage of the distribution of zeros of $f$.

Bibliographie

[1] Brelo M., Osnovy klassicheskoi teorii potentsiala, Mir, M., 1964 | MR | Zbl

[2] Brelo M., O topologiyakh i granitsakh v teorii potentsiala, Mir, M., 1974 | MR | Zbl

[3] Landkof N. S., Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966 | MR | Zbl

[4] Burbaki N., Integrirovanie mery, integrirovanie mer, Nauka, M., 1967 | MR

[5] Kheiman U., Kennedi P., Subgarmonicheskie funktsii, Mir, M., 1980

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

[7] Azarin V. S., “O luchakh vpolne regulyarnogo rosta tseloi funktsii”, Matem. sb., 79:4 (1969), 463–476 | MR | Zbl

[8] Ehrenpreis L., Malliavin P., “Invertible operators and interpolation in $\mathscr A\mathscr U$ spaces”, J. Math. pures et appl., 53 (1974), 165–182 | MR | Zbl

[9] Yulmukhametov R. S., “Asimptoticheskaya approksimatsiya subgarmonicheskikh funktsii”, Sib. matem. zhurn., 26:4 (1985), 159–175 | MR | Zbl

[10] Beurling A., Malliavin P., “On Fourier transforms of measures with compact support”, Acta Math., 107:3–4 (1962), 291–303 | DOI | MR

[11] Malliavin P., “On the multiplier theorem for Fourier transforms of measures with compact support”, Ark. for matem., 17:1 (1979), 69–81 | DOI | MR | Zbl

[12] Kondratyuk A. A., “Metod ryadov Fure dlya tselykh i meromorfnykh funktsii vpolne regulyarnogo rosta”, Matem. sb., 106:3 (1978), 386–408 | MR | Zbl

[13] Balashov S. K., “O funktsiyakh vpolne regulyarnogo rosta po krivym pravilnogo vrascheniya”, Izv. AN SSSR. Ser. matem., 40 (1976), 338–354 | Zbl

[14] Azarin V. S., “Ob asimptoticheskom povedenii subgarmonicheskikh funktsii konechnogo poryadka”, Matem. sb., 108:2 (1979), 147–167 | MR | Zbl

[15] Govorov N. V., Kraevaya zadacha Rimana s beskonechnym indeksom, Nauka, M., 1986 | MR | Zbl

[16] Grishin A. F., Sodin M. L., “Rost po luchu, raspredelenie kornei po argumentam tseloi funktsii konechnogo poryadka i odna teorema edinstvennosti”, Teoriya funktsii, funkts. analiz i ikh prilozh., 50, Vischa shkola, Kharkov, 1988, 47–61 | MR

[17] Levin B. Ya., Dopolneniya i ispravleniya k knige “Raspredelenie kornei tselykh funktsii”, Preprint, FTINT AN USSR, Kharkov, 1978

[18] Epifanov O. V., “K voprosu ob epimorfnosti operatora svertki v vypuklykh oblastyakh”, Matem. zametki, 1974, no. 3, 415–422 | MR | Zbl

[19] Tkachenko V. A., “Uravneniya tipa svertki v prostranstvakh analiticheskikh funktsionalov”, Izv. AN SSSR. Ser. matem., 41:2 (1977), 378–392 | MR | Zbl

[20] Garnett Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR | Zbl

[21] Evgrafov M. A., Asimptoticheskie otsenki i tselye funktsii, Nauka, M., 1979 | MR | Zbl

[22] Kelleher J., Taylor B. A., “Closed ideals in locally convex algebras of analytic functions”, J. Reine Angew. Math., 255 (1972), 190–209 | MR | Zbl

[23] Galdberg A. A., Ostrovskii I. V., Raspredelenie znachenii meromorfnykh funktsii, Nauka, M., 1970 | MR

[24] Brelot M., “Etude des fonctions sous-harmoniques au voisinage d'un point singulier”, Ann. Inst. Fourier, 1 (1949), 121–156 | MR | Zbl

[25] Kheifits A. I., “O subgarmonicheskikh funktsiyakh vpolne regulyarnogo rosta v poluploskosti”, DAN SSSR, 239:2 (1978), 282–285 | MR

[26] Dzhrbashyan M. M., Integralnye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti, Nauka, M., 1966

[27] Khabibullin B. N., “O malosti rosta na mnimoi osi tselykh funktsii eksponentsialnogo tipa s zadannymi nulyami”, Matem. zametki, 43:5 (1988), 644–650 | MR

[28] Kheifits A. I., “O kornyakh v poluploskosti tselykh funktsii tselogo poryadka”, Izv. SKNTs VSh, estestv. nauki, no. 4, Rostov-na-Donu, 1973, 116–117

[29] Azarin V. S., “O subgarmonicheskikh funktsiyakh vpolne regulyarnogo rosta v mnogomernom prostranstve”, DAN SSSR, 146:4 (1962), 743–746 | MR | Zbl

[30] Kalinin S. I., “K voprosu ob approksimatsii resheniya odnorodnogo uravneniya svertki posredstvom elementarnykh”, Matem. zametki, 32:2 (1982), 199–211 | MR | Zbl

[31] Agranovich P. Z., Logvinenko V. N., “Analog teoremy Valirona–Titchmarsha dlya dvuchlennykh asimptotik subgarmonicheskoi funktsii s massami na konechnoi sisteme luchei”, Sib. matem. zhurn., 24:5 (1985), 3–19 | MR

[32] Khabibullin B. N., Teoremy sravneniya i odnorodnosti dlya subgarmonicheskikh funktsii, Dis. $\dots$ kand. fiz.-matem. nauk, Rostov-na-Donu, 1985

[33] Khabibullin B. N., “Sravnenie subgarmonicheskikh funktsii po ikh assotsiirovannym meram”, Matem. sb., 125:4 (1984), 522–538 | MR | Zbl

[34] Yulmukhametov R. S., “Approksimatsiya subgarmonicheskikh funktsii”, Analysis Mathematica, 11:3 (1985), 257–282 | DOI | MR | Zbl