From Mendeleev to Fourier

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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

T(x) = a_0 + sum_{n=1}^N a_n cos nx + sum_{n=1}^N b_n sin nx

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.

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