I am not sure that this question is research level, but it was not answered at MSE for several days, so I place it here.

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.

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.