Bernstein inequality for trigonometric polynomials ($\Vert T_n^{\prime }{\Vert }_{\infty }\le n\Vert T_n{\Vert }_{\infty }$)

A trigonometric polynomial of degree at most $n$ obeys the displayed derivative bound. Iteration gives $\Vert T_n^{(r)}{\Vert }_{\infty }\le n^r\Vert T_n{\Vert }_{\infty }$. Applied to differences of approximants at dyadic degrees, this inequality underlies the inverse theorem for trigonometric approximation. It differs from the endpoint-weighted Bernstein inequality for algebraic polynomials.

Use the convention $K_m={\sum }_{|j|\le m}(1-|j|/(m+1))e^{ijt}$. Multiplying $K_{n-1}$ by $2n\sin (nt)$ gives Fourier coefficients $-ik$ for $|k|\le n$. A trigonometric polynomial of degree at most $n$ has no other frequencies, so convolution proves the displayed identity. Positivity and integral one of the Fejér kernel then give $\Vert P^{\prime }{\Vert }_{\infty }\le 2n\Vert P{\Vert }_{\infty }$. This is a useful elementary estimate; the sharp Bernstein inequality for trigonometric polynomials improves $2n$ to $n$.