|
|
| Line 1: |
Line 1: |
| In [[mathematical analysis]], '''Bernstein's inequality''' is named after [[Sergei Natanovich Bernstein]]. The inequality states that on the [[complex plane]], within the disk of radius 1, the degree of a [[polynomial]] times the maximum value of a polynomial is an upper bound for the similar maximum of its [[derivative]].
| | Royal Votaw is my name but I never really liked that name. To keep birds is one of the issues he enjoys most. Arizona is her birth location and she will never transfer. She functions as a financial officer and she will not alter it anytime quickly.<br><br>Review my web-site; extended car warranty ([http://Goodtogether.com/UserProfile/tabid/43/userId/63137/Default.aspx visit this hyperlink]) |
| | |
| ==Theorem==
| |
| Let ''P'' be a polynomial of degree <math>n</math> on complex numbers with derivative ''P′''. Then
| |
| | |
| :<math>\max_{|z| \le 1}( |P'(z)| ) \le n\cdot\max_{|z| \le 1}( |P(z)| ) </math>
| |
| | |
| The inequality finds uses in the field of [[approximation theory]].
| |
| | |
| Using the Bernstein's inequality we have for the ''k'':th derivative,
| |
| | |
| :<math>\max_{|z| \le 1}( |P^{(k)}(z)| ) \le \frac{n!}{(n-k)!} \cdot\max_{|z| \le 1}( |P(z)| ). </math>
| |
| | |
| ==See also==
| |
| | |
| * [[Markov brothers' inequality]]
| |
| * [[Remez inequality]]
| |
| | |
| ==References==
| |
| * {{cite journal | last=Frappier | first=Clément | title=Note on Bernstein's inequality for the third derivative of a polynomial | journal=J. Inequal. Pure Appl. Math. | volume=5 | number=1 | at=Paper No. 7 | year=2004 | issn=1443-5756 | url=http://www.emis.de/journals/JIPAM/images/154_03_JIPAM/154_03.pdf | zbl=1060.30003 }}
| |
| * {{cite book | last1=Rahman | first1=Q. I. | last2=Schmeisser | first2=G. | title=Analytic theory of polynomials | series=London Mathematical Society Monographs. New Series | volume=26 | location=Oxford | publisher=[[Oxford University Press]] | year=2002 | isbn=0-19-853493-0 | zbl=1072.30006 }}
| |
| | |
| [[Category:Inequalities]]
| |
| [[Category:Approximation theory]]
| |
| [[Category:Functional analysis]]
| |
| | |
| | |
| {{mathanalysis-stub}}
| |
| | |
| [[it:Disuguaglianza di Bernstein]]
| |
Revision as of 16:37, 25 February 2014
Royal Votaw is my name but I never really liked that name. To keep birds is one of the issues he enjoys most. Arizona is her birth location and she will never transfer. She functions as a financial officer and she will not alter it anytime quickly.
Review my web-site; extended car warranty (visit this hyperlink)