Article contents
Markov's and Bernstein's Inequalities on Disjoint Intervals
Published online by Cambridge University Press: 20 November 2018
Extract
In 1889, A. A. Markov proved the following inequality:
INEQUALITY 1. (Markov [4]). If pn is any algebraic polynomial of degree at most n then
where ‖ ‖A denotes the supremum norm on A.
In 1912, S. N. Bernstein established
INEQUALITY 2. (Bernstein [2]). If pn is any algebraic polynomial of degree at most n then
for x ∈ (a, b).
In this paper we extend these inequalities to sets of the form [a, b] ∪ [c, d]. Let Πn denote the set of algebraic polynomials with real coefficients of degree at most n.
THEOREM 1. Let a < b ≦ c < d and let pn ∈ Πn. Then
for x ∈ (a, b).
- Type
- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 1981
References
- 8
- Cited by