= Fejér-kernel differentiation identity
{c}
{title2=$P'=-P*(2nK_{n-1}(t)\sin(nt))$}
Use the convention $K_m=\sum_{|j|\leq m}(1-|j|/(m+1))e^{ijt}$. Multiplying $K_{n-1}$ by $2n\sin(nt)$ gives <Fourier coefficients> $-ik$ for $|k|\leq n$. 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 $\|P'\|_\infty\leq2n\|P\|_\infty$. This is a useful elementary estimate; the sharp <Bernstein inequality for trigonometric polynomials> improves $2n$ to $n$.
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