I am interested in a quantitative version of the principle that smoothness of a function on the real line is connected with decay of its Fourier transform. Namely, smoothness can be measured by the continuity modulus $\omega$: $$|f(x)-f(y)|\le \omega(|x-y|),$$ and for the decay of the Fourier transform of $f$ we consider estimates of the type $$|\hat f(x)|\le M(|x|).$$ Given $\omega$, for which the former estimate is fulfilled, what is $M$ in the latter estimate? And vice versa, given $M$, what is $\omega$?

I am interested in results where $\omega$ and $M$ are more or less arbitrary under the assumption that they are reasonably regular. I was not able to find an answer elsewhere.

  • 1
    $\begingroup$ There is the obvious $|f(x)-f(y)| = |\int_{-\infty}^\infty \hat{f}(\xi) (e^{2i \pi \xi x}-e^{2i \pi \xi y})d\xi|$ $ \le \int_{-\infty}^\infty |\hat{f}(\xi) (1-e^{2i \pi \xi (x-y)})|d\xi \le C |x-y|\int_{-\infty}^\infty |\hat{f}(\xi) \xi|d\xi$ $\endgroup$ – reuns Sep 15 '18 at 2:26

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