Unconditional exponential bases in Hilbert spaces

K. P. Isaev ; R. S. Yulmukhametov

Ufa mathematical journal, Tome 3 (2011) no. 1, pp. 3-15

Résumé

In the present paper, we consider the existence of unconditional exponential bases in general Hilbert spaces $H=H(E)$ consisting of functions defined on some set $E\subset\mathbb C$ and satisfying the following conditions. 1. The norm in the space $H$ is weaker than the uniform norm on $E$, i.e. the following estimate holds for some constant $A$ and for any function $f$ from $H$: $$ \|f\|_H\le A\sup_{z\in E}|f(z)|. $$ 2. The system of exponential functions $\{\exp(\lambda z),\lambda\in\mathbb C\}$ belongs to the subset $H$ and it is complete in $H$. It is proved that unconditional exponential bases cannot be constructed in $H$ unless a certain condition is carried out. Sufficiency of the weakened condition is proved for spaces defined more particularly.

Détail

Keywords: series of exponents, unconditional bases, Hilbert space.

Bibliographie

[1] Nikolskii N. K., Pavlov B. S., Khruschev S. V., Bezuslovnye bazisy iz eksponent i vosproizvodyaschikh yader. I, Preprint, LOMI, 8–80

[2] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965, 448 pp. | MR

[3] Nikolskii N. K., Lektsii ob operatore sdviga, Nauka, M., 1980 | MR

[4] N. Aronszajn, “Theory of reproducing kernels”, Transactions of the American Mathematical Society, 68:3 (1950), 337–404 | DOI | MR | Zbl

[5] Bashmakov R. A., Isaev K. P., Yulmukhametov R. S., “O geometricheskikh kharakteristikakh vypuklykh funktsii i integralax Laplasa”, Ufimskii mat. zhurn., 2:1 (2010), 3–16 | Zbl

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

[7] Isaev K. P., Yulmukhametov R. S., “Ob otsutstvii bezuslovnykh bazisov iz eksponent v prostranstvakh Bergmana na oblastyakh, ne yavlyayuschikhsya mnogougolnikami”, Izv. RAN. Ser. matem., 71:6 (2007), 69–90 | DOI | MR | Zbl

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

[9] Lutsenko V. I., Yulmukhametov R. S., “Obobschenie teoremy Peli–Vinera na vesovye prostranstva”, Matem. zametki, 48:5 (1990), 80–87 | MR | Zbl

[10] Lutsenko V. I., “Teorema Peli–Vinera na neogranichennom intervale”, Issledovaniya po teorii priblizhenii, Ufa, 1989, 79–85 | MR | Zbl

[11] Napalkov V. V., Bashmakov R. A., Yulmukhametov R. S., “Asimptoticheskoe povedenie integralov Laplasa i geometricheskie kharakteristiki vypuklykh funktsii”, DAN, 413:1 (2007), 20–22 | MR | Zbl

[12] Bashmakov R. A., Sistemy eksponent v vesovykh gilbertovkh prostranstvakh na $R$, Dissertatsiya na soiskanie uchenoi stepeni kandidata fiz.-mat. nauk, Institut matematiki s VTs UNTs RAN, 2006

[13] Sedletskii A. M., Klassy analiticheskikh preobrazovanii Fure i eksponentsialnye approksimatsii, M., 2005, 504 pp.

[14] Levin B. Ya., Lyubarskii Yu. I., “Interpolyatsiya tselymi funktsiyami spetsialnykh klassov i svyazannye s neyu razlozheniya v ryady eksponent”, Izv. AN SSSR. Ser. mat., 39:3 (1975), 657–702 | MR | Zbl

[15] Lyubarskii Yu. I., “Ryady eksponent v prostranstvakh Smirnova i interpolyatsiya tselymi funktsiyami spetsialnykh klassov”, Izv. AN SSSR. Ser. matem., 52:3 (1988), 559–580 | MR | Zbl

[16] Lutsenko V. I., Bezuslovnye bazisy iz eksponent v prostranstvakh Smirnova, Dissertatsiya na soiskanie uchenoi stepeni kandidata fiz.-mat. nauk, Institut matematiki s VTs UNTs RAN, 1992