Fourier Series Formalization in ACL2(r)

TL;DR

This paper formalizes key properties of Fourier series within ACL2(r), leveraging non-standard analysis to prove orthogonality, Fourier coefficients, and convergence theorems, advancing formal reasoning in Fourier analysis.

Contribution

It introduces a formal framework in ACL2(r) for Fourier series, including proofs of orthogonality, Fourier coefficients, and convergence theorems using non-standard analysis techniques.

Findings

Formal proof of orthogonality of trigonometric functions

Formalization of Fourier coefficient formulas

Proof of Dini Uniform Convergence Theorem

Abstract

We formalize some basic properties of Fourier series in the logic of ACL2(r), which is a variant of ACL2 that supports reasoning about the real and complex numbers by way of non-standard analysis. More specifically, we extend a framework for formally evaluating definite integrals of real-valued, continuous functions using the Second Fundamental Theorem of Calculus. Our extended framework is also applied to functions containing free arguments. Using this framework, we are able to prove the orthogonality relationships between trigonometric functions, which are the essential properties in Fourier series analysis. The sum rule for definite integrals of indexed sums is also formalized by applying the extended framework along with the First Fundamental Theorem of Calculus and the sum rule for differentiation. The Fourier coefficient formulas of periodic functions are then formalized from the orthogonality relations and the sum rule for integration. Consequently, the uniqueness of Fourier sums is a straightforward corollary. We also present our formalization of the sum rule for definite integrals of infinite series in ACL2(r). Part of this task is to prove the Dini Uniform Convergence Theorem and the continuity of a limit function under certain conditions. A key technique in our proofs of these theorems is to apply the overspill principle from non-standard analysis.