6
$\begingroup$

I'm sure that this is classical--but can anyone provide a reasonable example of an $L^\infty(\mathbb{T})$ function whose Fourier series is $\ell^2$ but no better? Not even $L^2\log L$? Presumably one exists but nothing has come to my mind. I'm trying to understand just how far one can stretch a (version of a) particular conjecture on Boolean functions, the Fourier Entropy-Influence conjecture.

$\endgroup$
  • 2
    $\begingroup$ I haven't got my copy of Katznelson's Introduction to Harmonic Analysis at hand, but that is one place I would look for information or an example $\endgroup$ – Yemon Choi Feb 8 '17 at 23:38
5
$\begingroup$

A theorem due to Kahane, Katznelson and de Leeuw says that for any sequence $(a_n)_{n\in \mathbb{Z}}$ belonging to $\ell_2$ there is a continuous function $f \in C(\mathbb{T})$ such that $|\widehat{f}(n)| \geqslant |a_n|$, so in general you don't get anything better than $\ell_2$. The proof can be found in Appendix B of Katznelson's book -- the construction is somewhat explicit.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.