# On a local Darlington synthesis problem

## TL;DR

This paper introduces a local version of the Darlington synthesis problem, relating algebraic embedding conditions to analytic pseudocontinuation, and proves a local analog of the Arov--Douglas--Helton theorem.

## Contribution

It formulates and proves a local version of the Darlington synthesis problem and extends the ADH theorem to a localized setting.

## Key findings

- Established a local Darlington synthesis framework
- Proved a local analog of the ADH theorem
- Connected algebraic and analytic properties in a localized context

## Abstract

The Darlington synthesis problem (in the scalar case) is the problem of embedding a given contractive analytic function to an inner $2\times 2$ matrix function as the entry. A fundamental result of Arov--Douglas--Helton relates this algebraic property to a pure analytic one known as a pseudocontinuation of bounded type. We suggest a local version of the Darlington synthesis problem and prove a local analog of the ADH theorem.