A note on the periodic Hilbert Transform on a strip

TL;DR

This paper proves a conjecture related to the finite depth water wave problem, using identities involving Jacobi Theta functions, and discusses potential implications of this mathematical improvement.

Contribution

It provides a proof of a conjecture by Constantin--Strauss--Ve2rve2ruce2, advancing understanding of the periodic Hilbert Transform on a strip.

Findings

Proved a conjecture related to water wave problems.

Utilized identities involving Jacobi Theta functions.

Discussed implications of the mathematical results.

Abstract

In this note we prove a conjecture by Constantin--Strauss--V\u{a}rv\u{a}ruc\u{a} related to the finite depth water wave problem, tightening their results. The proof uses identities related to Jacobi Theta functions. We also discuss potential implications of the improvement.