# Direct Cauchy Theorem and Fourier integral in Widom domains

## TL;DR

This paper explores Fourier integrals linked to the complex Martin function in Widom domains with the DCT property, impacting the analysis of reflectionless Weyl-Titchmarsh functions and related systems.

## Contribution

It introduces a Fourier integral framework in Widom domains with DCT and applies it to reflectionless functions and canonical systems, highlighting the importance of DCT.

## Key findings

- Fourier integral associated with the Martin function in Widom domains.
- DCT is crucial for the analysis of reflectionless Weyl-Titchmarsh functions.
- Application to canonical systems and transfer matrices.

## Abstract

We derive Fourier integral associated to the complex Martin function in the Denjoy domain of Widom type with the Direct Cauchy Theorem (DCT). As an application we study reflectionless Weyl-Titchmarsh functions in such domains, related to them canonical systems and transfer matrices. The DCT property appears to be crucial in many aspects of the underlying theory.