Refinement of the theory and convergence of the Sinc convolution -- beyond Stenger's conjecture

TL;DR

This paper refines the theoretical foundation of the Sinc convolution, resolving Stenger's conjecture, and enhances its convergence rate by adopting a double-exponential transformation, supported by theoretical and numerical evidence.

Contribution

It resolves key theoretical issues of the Sinc convolution, including Stenger's conjecture, and improves convergence speed through a double-exponential transformation.

Findings

Resolved Stenger's conjecture.

Achieved superior convergence with double-exponential transformation.

Validated improvements through numerical experiments.

Abstract

The Sinc convolution is an approximate formula for indefinite convolutions proposed by Stenger. The formula was derived based on the Sinc indefinite integration formula combined with the single-exponential transformation. Although its efficiency has been confirmed in various fields, several theoretical issues remain unresolved. The first contribution of this study is to resolve those issues by refining the underlying theory of the Sinc convolution. This contribution includes an essential resolution of Stenger's conjecture. The second contribution of this study is to improve the convergence rate by replacing the single-exponential transformation with the double-exponential transformation. Theoretical analysis and numerical experiments confirm that the modified formula achieves superior convergence compared to Stenger's original formula.