Relationship between max of a function and Fourier transform of that function

Question

I'm trying to see if any relationship can be inferred between the maximum of a function, and the Fourier transform of that function. Say I know the FT expression, can I infer the maximum in real domain?

I've been attempting several calculus but I'm wondering now if there's in fact nothing much to be said and if I'm wasting my time :/

I will post some calculation examples when i figure how to convert my mathtype into mathjax (gosh it is tedious). Apologies for now.

Thank you for any tip on this topic!

Answer 1

Here's a pretty easy relationship to derive. Suppose that $\hat{f}(k)$ is integrable. Then, $f(x)$ is continuous and bounded, and

$$ \max_{x\in\Bbb{R}}\vert f(x)\vert \leq C \int_{-\infty}^\infty\vert\hat{f}(k)\vert dk $$

To see why, just write the inverse Fourier transform (for a particular choice of definition):

$$ f(x) = \frac{1}{2\pi}\int_{-\infty}^\infty \hat{f}(k)\exp(ikx)dk $$

Then, by the triangle inequality,

$$ \vert f(x)\vert \leq \frac{1}{2\pi}\int_{-\infty}^\infty\vert \hat{f}(k)\vert dk $$