Akhiezer's theorem
In the mathematical field of complex analysis, Akhiezer's theorem is a result about entire functions proved by Naum Akhiezer.[1]
Akhiezer's theorem
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Dans le mathématique field of complex analysis, Akhiezer's theorem is a result about entire functions proved by Naum Akhiezer.[1]
Déclaration[]
Laisser F(z) be an entire function de exponential type t, avec F(X) ≥ 0 for real X. Then the following are equivalent:
- There exists an entire function F, de exponential type t/2, having all its zeros in the (fermé) upper half plane, tel que
!["{style]("https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8ad8ebc444c9cb2c87065e191a46513a2b4ca3")
- L'un a:
!["{displaystyle]("https://wikimedia.org/api/rest_v1/media/math/render/svg/74e4c073ef0bd4977ab5d522165a5be625c2ede1")
où zn are the zeros of F.
Résultats associés[]
It is not hard to show that the Fejér–Riesz theorem is a special case.[2]
Remarques[]
- ^ voir Akhiezer (1948).
- ^ voir Boas (1954) et Boas (1944) for references.
Références[]
Boas, Jr., Ralph Philip (1954), Entire functions, New York: Academic Press Inc., pp. 124–132
- Boas, Jr., R. P. (1944), "Functions of exponential type. I", Duc Math. J., 11: 9–15, est ce que je:10.1215/s0012-7094-44-01102-6, ISSN 0012-7094
- Akhiezer, N. je. (1948), "On the theory of entire functions of finite degree", Doklady Akademii Nauk SSSR, Nouvelle série, 63: 475–478, M 0027333
Si vous voulez connaître d'autres articles similaires à Akhiezer's theorem vous pouvez visiter la catégorie Théorèmes en analyse complexe.
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