Indefinite Integration Operators Identities and their Approximations

TL;DR

This paper introduces new identities for indefinite integration operators and demonstrates their use in developing highly accurate approximation methods for integral equations, convolutions, and transform inversions, with applications in statistical modeling.

Contribution

The paper presents novel identities for indefinite integration operators and develops efficient approximation techniques, notably using Legendre polynomials, for various integral-related problems.

Findings

Accurate approximation of indefinite integrals with only 5 sample points.

Reconstruction of statistical models with nearly 3 significant figures accuracy.

Enhanced methods for solving Wiener--Hopf equations and related integral transforms.

Abstract

The integration operators (*) $({\mathcal J}^+\,g)(x) = \int_a^x g(t) \, dt$ and (**) $({\mathcal J}^-\,g)(x) = \int_x^b g(t) \, dt$ defined on an interval $(a,b) \subseteq {\mathbf R}$ yield new identities for indefinite convolutions, control theory, Laplace and Fourier transform inversion, solution of differential equations, and solution of the classical Wiener--Hopf integral equations. These identities are are expressed in terms of ${\mathcal J}^\pm$\, and they are thus esoteric. However the integrals (*) and (**) can be approximated in many ways, yielding novel and very accurate methods of approximating all of the above listed relations. Several examples are presented, mainly using Legendre polynomial as approximations, and references are given for approximation of some of the operations using Sinc methods. These examples illustrate for a class of sampled statistical models, the possibility of reconstructing models much more efficiently than by the usual slow Monte--Carlo $({\mathcal O}(N^{-1/2})$ rate. Our examples illustrate that we need only sample at $5$ points to get a representation of a model that is uniformly accurate to nearly $3$ significant figure accuracy.