Complex Analysis of Real Functions I: Complex-Analytic Structure and Integrable Real Functions

TL;DR

This paper introduces a complex-analytic framework within the unit disk that allows representing and analyzing a broad class of real functions, including those with discontinuities or unbounded behavior, via restrictions of analytic functions.

Contribution

It provides an explicit method to construct analytic functions from real functions, establishing a universal complex-analytic structure as a regulator for real function analysis.

Findings

Any integrable real function can be represented by an analytic function on the unit circle.

The framework includes non-differentiable, discontinuous, and unbounded functions.

A constructive approach to derive analytic functions from real functions is presented.

Abstract

A complex-analytic structure within the unit disk of the complex plane is presented. It can be used to represent and analyze a large class of real functions. It is shown that any integrable real function can be obtained by means of the restriction of an analytic function to the unit circle, including functions which are non-differentiable, discontinuous or unbounded. An explicit construction of the analytic functions from the corresponding real functions is given. The complex-analytic structure can be understood as an universal regulator for analytic operations on real functions.