Second-Order Parameterizations for the Complexity Theory of Integrable Functions

2025-06-12

AI Summary

This paper establishes a unified second-order parameterized complexity theory for the Banach space $L^p([0,1];mathbb{C})$, where $1 leq p < infty$. It addresses three natural parameterizations in this space: the binary $L^p$-norm, the convergence rate of Fourier series, and the approximation rate by step functions. The work proves, for the first time, their mutual linear equivalence. Methodologically, it extends second-order computability theory from spaces of continuous functions to integrable function spaces, integrating tools from computable analysis, Fourier analysis, real analysis, and approximation theory to construct a rigorous formal framework. The main contribution is the establishment of equivalence among these three analytical parameters—thereby filling a foundational gap in the theory of computational complexity for integrable functions, which previously lacked a unified, tight, and computably grounded characterization. This result provides the first parameterized foundation for algorithm design and complexity analysis applicable to $L^p$ spaces.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Learning Theory
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• Develop second-order parameterized complexity theory for integrable functions
• Generalize complexity theory from continuous to integrable function spaces
• Prove equivalence of three parameterizations for Lp space

Innovation

• Unified second-order parameterized complexity theory
• Generalizes complexity theory for continuous functions
• Proves equivalence of three natural parameterizations