From Mendeleev to Fourier

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 degreen, and |P(x)| 1 on [-1, 1] then |P'(x)| n^2 on [-1, 1].

IfP(x) is a trigonometric polynomial then Bernstein proved that the bound decreases from n^2 ton.

Theorem (Bernstein): If P(x) is a trigonometric polynomial of degree n, and |P(z)| 1 on [-, ] then |P'(x)| n on [-, ].

Now a trigonometric polynomial is a truncated Fourier series

and so the max norm of the T'is no more thann times the max norm of T.

This post and the previous one were motivated by Terence Tao's latest post on Bernstein theory.

Related posts

• Zeros of trig polynomials
• Convert between real and complex Fourier series
• Systematically solving trig equations

The post From Mendeleev to Fourier first appeared on John D. Cook.