On Integral Theorems and their Statistical Properties

TL;DR

This paper introduces a new class of integral theorems based on cyclic functions, providing natural density estimators via Monte Carlo methods, with a focus on optimizing cyclic functions for improved estimation accuracy.

Contribution

It develops a novel framework of integral theorems using cyclic functions and variational methods, extending Fourier integral theorems and proposing optimal cyclic functions for density estimation.

Findings

New class of integral theorems based on cyclic functions

Density estimators derived from Monte Carlo methods

Identification of optimal cyclic functions minimizing square integrals

Abstract

We introduce a class of integral theorems based on cyclic functions and Riemann sums approximating integrals. The Fourier integral theorem, derived as a combination of a transform and inverse transform, arises as a special case. The integral theorems provide natural estimators of density functions via Monte Carlo methods. Assessments of the quality of the density estimators can be used to obtain optimal cyclic functions, alternatives to the sin function, which minimize square integrals. Our proof techniques rely on a variational approach in ordinary differential equations and the Cauchy residue theorem in complex analysis.