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

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.