Comparing representations for function spaces in computable analysis

TL;DR

This paper analyzes various representations of function spaces in computable analysis, comparing their computational complexity using Weihrauch reducibility, with a focus on analytic functions, polynomials, and smooth functions.

Contribution

It provides a systematic comparison of different representations of function spaces and characterizes their Weihrauch degrees, revealing the computational complexity of translating between them.

Findings

Non-computable translations are Weihrauch equivalent to closed choice on natural numbers.

Representations of polynomials involve the Weihrauch degree LPO* for zero-finding.

Smooth functions and Schwartz functions relate to closed choice and lim degrees.

Abstract

This paper compares different representations (in the sense of computable analysis) of a number of function spaces that are of interest in analysis. In particular subspace representations inherited from a larger function space are compared to more natural representations for these spaces. The formal framework for the comparisons is provided by Weihrauch reducibility. The centrepiece of the paper considers several representations of the analytic functions on the unit disk and their mutual translations. All translations that are not already computable are shown to be Weihrauch equivalent to closed choice on the natural numbers. Subsequently some similar considerations are carried out for representations of polynomials. In this case in addition to closed choice the Weihrauch degree LPO* shows up as the difficulty of finding the degree or the zeros. As a final example, the smooth functions are contrasted with functions with bounded support and Schwartz functions. Here closed choice on the natural numbers and the lim degree appear.