Inequality for an integrable real valued function with a compactly supported Fourier transform

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 outright constant $c$.

2022-06-07 14:32:11
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Answers: 1

Write Fourier inversion formula for $f(x)$ and also $f(0)$. You get an expression of $f(x)-f(0)$ as an indispensable on the portable set $[-\gamma, \gamma]$. After that you simply need to bound all the terms in the indispensable (to bound $1-e^{iyx}$, you might intend to share it as an indispensable).

2022-06-07 14:58:36