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.