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Let $f$ be an integrable function on $\mathbb{R}$ where $\operatorname{support}(\widehat{f}) \subseteq [-\gamma, \gamma]$ for some $ 0 < \gamma < 1$

Prove that $\lvert f(x) - f(0)\rvert \leq c \gamma \lvert x\rvert \sup\limits_{ y \in \mathbb{R}}\left\{(1+|y|)\lvert f(y)\rvert\right\}$ for some absolute constant $c$.

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Write Fourier inversion formula for $f(x)$ and $f(0)$. You get an expression of $f(x)-f(0)$ as an integral on the compact set $[-\gamma, \gamma]$. Then you just have to bound all the terms in the integral (to bound $1-e^{iyx}$, you may want to express it as an integral).

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    The $|x|$ appears when you write $|\int_0^x e^{ity}| \le |x|$.2011-05-22