The previous post looked at an inequality discovered by Dmitri Mendeleev and generalized by Andrey Markov:
Theorem (Markov): If P(x) is a real polynomial of degree n, and |P(x)| ≤ 1 on [−1, 1] then |P′(x)| ≤ n² on [−1, 1].
If P(x) is a trigonometric polynomial then Bernstein proved that the bound decreases from n² to n.
Theorem (Bernstein): If P(x) is a trigonometric polynomial of degree n, and |P(z)| ≤ 1 on [−π, π] then |P′(x)| ≤ n on [−π, π].
Now a trigonometric polynomial is a truncated Fourier series
and so the max norm of the T′ is no more than n times the max norm of T.
This post and the previous one were motivated by Terence Tao’s latest post on Bernstein theory.