2
$\begingroup$

Let $H$ be the Hilbert transform. Is there a continuous, even function $h:\mathbb{R}\to \mathbb{R}$ with support on $[-1,1]$ such that, for some $\lambda\in \mathbb{R}$, $$H(h')(t) = \lambda h(t)$$ for all $-1< t< 1$?

What is the set of all such functions?

An answer would complete the solution to Distribution $f$ such that (a) $\widehat{f}$ has compact support, (b) $\mathbb{E}(|X|)$ is minimal? , which is itself of use for several problems related to $\zeta(s)$ and $L$-functions.

Improve this question
$\endgroup$
16
  • $\begingroup$ I can probably work this out as a power series $\sum_{n=0}^\infty a_n x^{2 n}$ vanishing at $x=1$, but I feel like this problem must have been considered before. $\endgroup$
    – H A Helfgott
    Commented Nov 2, 2022 at 17:00
  • $\begingroup$ Perhaps there is a simple argument showing what the avaluable value(s) of $\lambda$ are? $\endgroup$
    – H A Helfgott
    Commented Nov 2, 2022 at 17:03
  • $\begingroup$ (Am I right in guessing that there is a solution $h$ only for $\lambda=1$?) $\endgroup$
    – H A Helfgott
    Commented Nov 2, 2022 at 17:15
  • 1
    $\begingroup$ If I understand correctly, then $H h' = (-\Delta)^{1/2} h$, where $\Delta u = u''$ is the 1-D Laplace operator, and $h$ is just an eigenfunction of $(-\Delta)^{1/2}$ in $(-1,1)$ with zero exterior condition. There does not seem to be an explicit expression for these eigenfunctions, but they have been studied quite a lot. The minimal $\lambda$ is $1.157773\ldots$; see, for example, my paper with Bartłomiej Dyda and Alexey Kuznetsov, DOI:10.1112/jlms.12024. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Nov 2, 2022 at 19:49
  • $\begingroup$ @MateuszKwaśnicki Is there perhaps a nice expression for $h\ast h$? $\endgroup$
    – H A Helfgott
    Commented Nov 2, 2022 at 20:15

0

Reset to default

You must log in to answer this question.