1
$\begingroup$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a Schwartz function. Suppose that $\left|\hat{f}(\omega)\right|\leq1$, $\left|\hat{f}(\omega)\right|\leq\left|\omega\right|^{-4}$. Show that: $ \left|f(3)-f(1)\right|<1000$

One thing is that Schwartz functions are invariant under the Fourier transform $\mathcal{F}$. Essentially, we can write $f(x)$ as: $$ f(x)=\frac{1}{2\pi}\int_{\mathbb{R}}e^{ixt}\hat{f}(t)dt$$

I am having some difficulties dealing with the estimates after that.

  • 0
    Hint: $\hat f$ is integrable. (And I get much less than 1000 in the RHS.)2011-08-26

3 Answers 3