Comparing Representations for Function Spaces in Computable Analysis

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.