Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Local Behavior of Entire Functions of Exponential Type

Local Behavior of Entire Functions of Exponential Type We consider entire functions of exponential type τ > 0, whose modulus is bounded by a constant M at the extrema of sin(τz), and which vanish at the origin. Extending a result of L. Hörmander, we show that if f is any such function, then ¦f(x)¦ ≤ M¦ sin(τx)¦ for all x ∈ (−π/(2τ),π/(2τ)), provided that f(x) = o(x) as x → ± ∞; furthermore, equality holds at any point x ∈ (−π/(2τ), 0) ∪ (0, π/(2τ)) if and only if f(z) ≡ e iγ sin(τz) for some γ ∈ R. This also generalizes a result due to R. P. Boas Jr. about trigonometric polynomials. Besides, we prove some other results for entire functions of order 1 and type τ > 0, one being an analogue of a result of M. Riesz about trigonometric polynomials whose zeros are all real and simple. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Local Behavior of Entire Functions of Exponential Type

Download PDF
18 pages

Loading next page...
 
/lp/springer-journals/local-behavior-of-entire-functions-of-exponential-type-37orG40WU0

References (17)

Publisher
Springer Journals
Copyright
Copyright © 2002 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321852
Publisher site
See Article on Publisher Site

Abstract

We consider entire functions of exponential type τ > 0, whose modulus is bounded by a constant M at the extrema of sin(τz), and which vanish at the origin. Extending a result of L. Hörmander, we show that if f is any such function, then ¦f(x)¦ ≤ M¦ sin(τx)¦ for all x ∈ (−π/(2τ),π/(2τ)), provided that f(x) = o(x) as x → ± ∞; furthermore, equality holds at any point x ∈ (−π/(2τ), 0) ∪ (0, π/(2τ)) if and only if f(z) ≡ e iγ sin(τz) for some γ ∈ R. This also generalizes a result due to R. P. Boas Jr. about trigonometric polynomials. Besides, we prove some other results for entire functions of order 1 and type τ > 0, one being an analogue of a result of M. Riesz about trigonometric polynomials whose zeros are all real and simple.

Journal

Computational Methods and Function Theory – Springer Journals

Published: Apr 2, 2013

There are no references for this article.