An approximation theorem for entire functions of

B. N. Khabibullin

Sbornik. Mathematics, Tome 195 (2004) no. 1, pp. 135-148

Résumé

Let $L$ be an entire function of exponential type in $\mathbb C$ with indicator function $h_L$; let $\Lambda=\{\lambda_n\}$, $n=1,2,\dots$, be a subsequence of zeros of the entire function of exponential type $L\not\equiv0$; let $\Gamma=\{\gamma_n\}$ be a complex number sequence and assume that $$ \sum_n\biggl|\frac1{\lambda_n}-\frac1{\gamma_n}\biggr|<\infty. $$ A simple construction of a sequence of entire functions of exponential type $\{L_n\}$ transforming $\Lambda$ into a subsequence $\Gamma$ of zeros of an entire function of exponential type $G\not\equiv0$ such that $h_G=h_L$ is put forward (an approximation theorem). This result is applied to stability problems of zero sequences and non-uniqueness sequences for spaces of entire functions of exponential type with constraints on the indicators and to the problem of the stability of the completeness property of exponential systems in the space of germs of analytic functions on a compact convex set.

Bibliographie

[1] Levin B. Ya., Raspredelenie kornei tselykh funktsii, Fizmatgiz, M., 1956

[2] Levin B. Ya., Lectures on entire functions, Transl. Math. Monogr., 150, Amer. Math. Soc., Providence, RI, 1996 | MR | Zbl

[3] Krasichkov I. F., “Sravnenie tselykh funktsii konechnogo poryadka po raspredeleniyu ikh kornei”, Matem. sb., 70(112):2 (1966), 198–230 | MR | Zbl

[4] Krasichkov-Ternovskii I. F., “Sravnenie tselykh funktsii tselogo poryadka po raspredeleniyam ikh kornei”, Matem. sb., 71(113):3 (1966), 405–419 | MR

[5] Grishin A. F., “O regulyarnosti rosta subgarmonicheskikh funktsii. I–II”, Teoriya funktsii, funkts. analiz i ikh prilozh., 6 (1968), 3–29 | MR | Zbl

[6] Azarin V. S., “O luchakh vpolne regulyarnogo rosta tseloi funktsii”, Matem. sb., 79(121):4 (1969), 463–476 | MR | Zbl

[7] Azarin V. S., “Ob asimptoticheskom povedenii subgarmonicheskikh funktsii konechnogo poryadka”, Matem. sb., 108 (150):2 (1979), 147–167 | MR | Zbl

[8] Khabibullin B. N., “Sravnenie subgarmonicheskikh funktsii po ikh assotsiirovannym meram”, Matem. sb., 125 (167):4 (1984), 522–538 | MR | Zbl

[9] Khabibulin B. N., “Razlozhenie tselykh funktsii konechnogo poryadka na ekvivalentnye mnozhiteli”, Voprosy approksimatsii funktsii veschestvennogo i kompleksnogo peremennykh, Izd-vo Bashkirskogo otd. AN SSSR, Ufa, 1983, 161–181 | MR

[10] Napalkov V. V., Solomesch M. I., “Otsenka izmeneniya tseloi funktsii pri sdvigakh ee nulei”, Dokl. RAN, 342:6 (1995), 739–741 | MR | Zbl

[11] Hayman W. K., Subharmonic functions, vol. II, Academic Press, London, 1989 | MR | Zbl

[12] Khabibullin B. N., “Nekonstruktivnye dokazatelstva teoremy Berlinga–Malyavena o radiuse polnoty i teoremy needinstvennosti dlya tselykh funktsii”, Izv. RAN. Ser. matem., 58:4 (1994), 125–148 | Zbl

[13] Köthe G., “Dualität in der Funktionentheorie”, J. Reine Angew. Math., 191:1–2 (1953), 30–49 | MR | Zbl

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

[15] Ibragimov I. I., Metody interpolyatsii funktsii i nekotorye ikh primeneniya, Nauka, M., 1971 | MR

[16] Sedletskii A. M., Fourier transforms and approximations, GBS Publ, Amsterdam, 2000 | MR

[17] Khabibullin B. N., “Polnota sistem tselykh funktsii v prostranstvakh golomorfnykh funktsii”, Matem. zametki, 66:4 (1999), 603–616 | MR

[18] Khabibullin B. N., “Completeness of sets of complex exponentials in convex sets”, Trudy mezhdunarodnoi konferentsii “Kompleksnyi analiz, differentsialnye uravneniya i smezhnye voprosy”. III. Analiz i differentsialnye uravneniya (Ufa, iyun, 2000), IM s VTs UNTs RAN, Ufa, 2000, 56–63

[19] Khabibullin B. N., “Closeness of subharmonic and entire functions, stability of completeness of exponential systems, spectral synthesis”, Second International Conference “Mathematical Analysis and Economics” (Sumy, April, 2003), UAB, KKNU, VILTPE of NASU, IM of NASU, Sumy–Kharkiv–Kiev, 2003, 24–25

[20] Redheffer R. M., Private communication, May 2001

[21] Titchmarsh E., Teoriya funktsii, Nauka, M., 1980 | MR | Zbl

[22] Edvards R., Funktsionalnyi analiz, Mir, M., 1969

[23] Redheffer R. M., “Two consequences of the Beurling–Malliavin theory”, Proc. Amer. Math. Soc., 16:1 (1972), 116–122 | DOI | MR

[24] Redheffer R. M., “Completeness of sets of complex exponentials”, Adv. Math., 24 (1977), 1–62 | DOI | MR | Zbl

[25] Beurling A., Malliavin P., “On the closure of characters and the zeros of entire functions”, Acta Math., 118 (1967), 79–93 | DOI | MR | Zbl

[26] Beurling A., Malliavin P., “On Fourier transforms of measures with compact support”, Acta Math., 107 (1962), 291–309 | DOI | MR | Zbl

[27] Koosis P., The logarithmic integral, vol. I, II, Cambridge Univ. Press, Cambridge, 1988, 1992 | Zbl

[28] Koosis P., Leçons sur le théorème de Beurling et Malliavin, Les Publications CRM, Montréal, 1996 | MR | Zbl

[29] Havin V. P., Jöricke B., The uncertainty principle in harmonic analysis, Springer-Verlag, Berlin, 1994 | MR

[30] Redheffer R. M., “A note on completeness”, Notices Amer. Math. Soc., 16 (1967), 830

[31] Redheffer R. M., “Ganze Funktionen und Vollständigkeit”, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber., 6 (1957), 96–99