# On Bernstein’s inequality for polynomials

On Bernstein’s inequality for polynomials Bernstein’s classical inequality asserts that given a trigonometric polynomial T of degree $$n\ge 1$$ n ≥ 1 , the sup-norm of the derivative of T does not exceed n times the sup-norm of T. We present various approaches to prove this inequality and some of its natural extensions/variants, especially when it comes to replacing the sup-norm with the $$L^p-{\textit{norm}}$$ L p - norm . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# On Bernstein’s inequality for polynomials

, Volume 9 (3) – Mar 20, 2019
27 pages

/lp/springer-journals/on-bernstein-s-inequality-for-polynomials-3VTBKiHgPl
Publisher
Springer Journals
Copyright © 2019 by Springer Nature Switzerland AG
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-019-00294-x
Publisher site
See Article on Publisher Site

### Abstract

Bernstein’s classical inequality asserts that given a trigonometric polynomial T of degree $$n\ge 1$$ n ≥ 1 , the sup-norm of the derivative of T does not exceed n times the sup-norm of T. We present various approaches to prove this inequality and some of its natural extensions/variants, especially when it comes to replacing the sup-norm with the $$L^p-{\textit{norm}}$$ L p - norm .

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Mar 20, 2019