Second-Order Parameterizations for the Complexity Theory of Integrable Functions

TL;DR

This paper introduces a second-order parameterized complexity framework for integrable functions, extending existing theories for continuous functions, and establishes equivalences among different parameterizations of $L^p$ spaces.

Contribution

It generalizes second-order parameterized complexity theory to $L^p$ spaces and proves the mutual linear equivalence of three natural parameterizations.

Findings

Proves mutual linear equivalence of three parameterizations of $L^p$ spaces.

Extends complexity theory from continuous to integrable functions.

Provides a unified framework for analyzing integrable functions' complexity.

Abstract

We develop a unified second-order parameterized complexity theory for spaces of integrable functions. This generalizes the well-established case of second-order parameterized complexity theory for spaces of continuous functions. Specifically we prove the mutual linear equivalence of three natural parameterizations of the space $\Lrm{p}$ of $p$-integrable complex functions on the real unit interval: (binary) $\Lrm{p}$-modulus, rate of convergence of Fourier series, and rate of approximation by step functions.