Computable Jordan Decomposition of Linear Continuous Functionals on $C[0;1]$

TL;DR

This paper extends the computable Riesz representation theorem to show that Jordan decomposition of linear continuous functionals on C[0,1] is computable, linking functionals, measures, and functions of bounded variation.

Contribution

It establishes the computability of the Jordan decomposition across spaces of functionals, measures, and functions, extending prior results in computable analysis.

Findings

Computable bijections between the spaces are established.

Jordan decomposition is shown to be computable in each space.

The study introduces natural representations for these spaces.

Abstract

By the Riesz representation theorem using the Riemann-Stieltjes integral, linear continuous functionals on the set of continuous functions from the unit interval into the reals can either be characterized by functions of bounded variation from the unit interval into the reals, or by signed measures on the Borel-subsets. Each of these objects has an (even minimal) Jordan decomposition into non-negative or non-decreasing objects. Using the representation approach to computable analysis, a computable version of the Riesz representation theorem has been proved by Jafarikhah, Lu and Weihrauch. In this article we extend this result. We study the computable relation between three Banach spaces, the space of linear continuous functionals with operator norm, the space of (normalized) functions of bounded variation with total variation norm, and the space of bounded signed Borel measures with variation norm. We introduce natural representations for defining computability. We prove that the canonical linear bijections between these spaces and their inverses are computable. We also prove that Jordan decomposition is computable on each of these spaces.