On a generalization of an inequality of Bohr

B. F. Ivanov

Problemy analiza, Tome 2 (2013) no. 2, pp. 21-58.

Résumé

Let $p\in (1, 2], n\ge 1, S\subseteq R^{n}$ and $\Gamma(S, p)$— the set of all functions, $\gamma(t)\in L ^{p}(R ^{n})$ the support of the Fourier transform of which lies in $S$. We obtain the inequality conditions $||\int \limits_{E_t}\gamma(\tau)d\tau|| _{L ^{\infty}(R^n)}\le C||\gamma(\tau)|| _{L ^{p}(R^n)}$, where $t=(t _{1}, t _{2}, \dots , t _{n})\in R^{n}, E _{t} = \{\tau|\tau=(\tau _{1},\tau _{2},\dots ,\tau _{n})\in R^{n}, \tau_j\in [0,t_j]$, if $ t_j\ge 0$ and $\tau_{j}\in (t_j,0]$, if $\tau_{j} 0, 1\le j\le n\}, \gamma(\tau)\in \Gamma(S,p)$ and constant $C$ does not depend on $\gamma(t)$. Also were considered some validity conditions on the inequality on non-trivial subsets $\Gamma(S, p)$ in cases, where they were not satisfied on the whole $\Gamma(S, p)$.

Détail

Keywords: inequality of Bohr.

Bibliographie

[1] Bohr H., Un théoréme général sur l'intégration d'un polynome trigonométrigue, Comptes Rendus De L'Academie des sciences, 200, no. 15, 1935

[2] Bohr H., “Ein allgemeiner Sats über die Integration eines trigonometrischen Polynomials”, Prace Mathematyzcne Fizyczne, 43 (1935), 273–288; Collected Mathematical works, v. 2, 1952, 36

[3] Favard J., “Sur une propriete extremale de l'integrale d'une function periodique”, Cjmptes Rendus De L'Academie des Sciences, 202 (1936), 273–276

[4] Favard J., “Application de la formule sommatorie d'Euler á la demonstration de quelques propriétés extremales des integrales des fonctions periodiques on Presque-periodiques”, Matematisk Tidsskrift, 1936, 81–94

[5] Levitan B. M., “Ob odnom obobschenii neravenstv S. N. Bernshteina i N. Bohr'a”, DAN SSSR., XV:4 (1937), 169–172 | MR

[6] Hormander L., “A new proof and a generalization of an inequality of Bohr”, Mathematica Scandinavica, 2 (1954), 33–45 | MR | Zbl

[7] Kamzolov A. I., “O neravenstve Bora-Favara dlya funktsii na kompaktnykh simmetricheskikh prostranstvakh ranga I”, Matematicheskie zametki, 33:2 (1983), 187–193 | MR | Zbl

[8] Akopyan R. R., “Neravenstva Bora i Bernshteina dlya analiticheskikh i ogranichennykh v poluploskosti funktsii”, Materialy mezhdunarodnoi nauchnoi konferentsii “Sovremennye problemy matematiki, mekhaniki, informatiki” (Rossiya. Tula 28–30 noyabrya 2006g.)

[9] Baskakov A. G., SintyaevaK. A., “O neravenstvakh Bora-Favara dlya operatorov”, Izvestiya vuzov. Matematika, 2009, no. 12, 14–21 | MR | Zbl

[10] Yudin V. A. K neravenstvu Bora, Trudy instituta matematiki i mekhaniki Uralskogo otdeleniya RAN, 16:4 (2010), 312–313

[11] Kuptsov N. P., “Pryamye i obratnye teoremy teorii priblizhenii i polugruppy operatorov”, Uspekhi matematicheskikh nauk, XXIII:4(142) (1968), 118–178

[12] Bredikhina E. A., “O priblizhenii pochti-periodicheskikh funtsii s ogranichennym spektrom”, Matematicheskii sbornik, 56(98):1 (1962), 59–76 | Zbl

[13] Ivanov B. F., “Chastotnyi kriterii ogranichennosti reshenii odnogo klassa lineinykh sistem”, Differentsialnye uravneniya, 33:5 (1997), 704–706 | MR | Zbl

[14] Ivanov B. F., “Chastotnyi kriterii gladkosti po parametram reshenii odnogo klassa lineinykh sistem”, Differentsialnye uravneniya, 33:7 (1997), 1001 | MR | Zbl

[15] Funktsionalnyi analiz, Spravochnaya matematicheskaya biblioteka, ed. S. G. Kreina, Nauka, M., 1972

[16] Kolmogorov A.N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1968 | MR | Zbl

[17] Titchmarsh E., Vvedenie v teoriyu integralov Fure, Gostekhizdat, M. L., 1948

[18] Akhiezer N. I., Lektsii po teorii approksimatsii, Nauka, M., 1965 | MR

[19] Burbaki N., Integrirovanie. Mery, integrirovanie mer, Nauka, M., 1967 | MR