If $$ D = \sum_{k=-N}^{N}e^{ik\theta} = \frac{\sin((N + \frac{1}{2})\theta)}{\sin(\theta/2)} $$I want to show that $$|D_{N}(\theta)|\geq c \frac{\sin((N + \frac{1}{2})\theta)}{|\theta|}$$Could anyone give me a hint? I am stucked.
2026-05-04 15:55:38.1777910138
Problem 2. in Stein & Shakarachi (Dirichlet kernel violates the second property of good kernels)
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So it suffices to show that $| \theta | > c\sin (\frac{\theta}2)$? Do you know how to show $x \ge \sin x$ based on calculus? (take derivative of $f(x) = x - \sin x$ etc.)