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

Tomoaki Okayama

TL;DR

This work revises the Sinc convolution framework for indefinite convolutions by relaxing the previous analyticity and spectral assumptions, thereby enabling well-defined $F(\mathcal{J}_m^{\text{SE}})$ without requiring Stenger's conjecture. It introduces a double-exponential (DE) transformation, achieving substantially faster convergence than the original SE scheme while preserving rigorous resolvent-based error analysis. Convergence theorems are established for both SE and DE formulations under refined assumptions, with explicit rates: root-exponential for SE and nearly exponential for DE, validated by extensive numerical tests. The results provide a practically robust and theoretically sound approach for high-accuracy indefinite convolutions, with potential extensions to unbounded domains and broader function classes.

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.

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

TL;DR

This work revises the Sinc convolution framework for indefinite convolutions by relaxing the previous analyticity and spectral assumptions, thereby enabling well-defined without requiring Stenger's conjecture. It introduces a double-exponential (DE) transformation, achieving substantially faster convergence than the original SE scheme while preserving rigorous resolvent-based error analysis. Convergence theorems are established for both SE and DE formulations under refined assumptions, with explicit rates: root-exponential for SE and nearly exponential for DE, validated by extensive numerical tests. The results provide a practically robust and theoretically sound approach for high-accuracy indefinite convolutions, with potential extensions to unbounded domains and broader function classes.

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.

Paper Structure

This paper contains 24 sections, 35 theorems, 160 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Sinc approximation and Sinc indefinite integration
  3. Original and mat-vec friendly SE-Sinc indefinite integration
  4. Original and mat-vec friendly DE-Sinc indefinite integration
  5. Rewriting the indefinite convolution
  6. Existence of the resolvent of J
  7. Derivation of the expression of p(x) using the "Laplace transform"
  8. Approximate formulas for the indefinite convolution and their convergence theorems (main results)
  9. Sinc convolution combined with the SE transformation
  10. Sinc convolution combined with the DE transformation
  11. Existence of the resolvent of approximated operators
  12. Existence of the resolvent of JmSE
  13. Existence of the resolvent of JmDE
  14. Analysis of matrices appearing in the Sinc convolution
  15. Analysis in the case of the SE transformation
  16. ...and 9 more sections

Key Result

Theorem 2.1

Assume that $g$ is analytic on $\psi^{\text{\scriptsize{\rm SE}}}(\mathscr{D}_d)$ for $d\in(0,\uppi)$, and that there exist positive constants $K$, $\alpha$ and $\beta$ such that holds for all $z\in\psi^{\text{\scriptsize{\rm SE}}}(\mathscr{D}_d)$. Let $\mu = \min\{\alpha, \beta\}$, $n$ be a positive integer, and $h$ be selected by the formula Moreover, let $M$ and $N$ be positive integers defin

Figures (10)

  • Figure 1: Contours $C_{R}$, $C_{R,\gamma}^{+}$, $C_{R,\gamma}^{-}$, and $B_{R,\gamma}$.
  • Figure 2: Numerical results for Example \ref{['ex:1']}.
  • Figure 3: Numerical results for Example \ref{['ex:2']}.
  • Figure 4: Numerical results for Example \ref{['ex:3']}.
  • Figure 5: Numerical results for Example \ref{['ex:4']}.
  • ...and 5 more figures

Theorems & Definitions (57)

  • Theorem 2.1: Okayama-et-al
  • Theorem 2.2: stenger95:_colloc
  • Theorem 2.3: Okayama-et-al
  • Theorem 2.4: okayama22:_yet_de_sinc
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Definition 1
  • Lemma 3.3: Okayama et al. okayama15:_theor_sinc_nystr_volter
  • ...and 47 more