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Improvement of conformal maps combined with the Sinc approximation for derivatives over infinite intervals

Tomoaki Okayama, Yuito Kuwashita, Ao Kondo

TL;DR

The paper targets efficient numerical differentiation over infinite intervals by enhancing Stenger's Sinc-approximation-based formulas with improved conformal maps. By replacing the i=2 and i=4 maps with $\phi_2(x)=\log(1+e^x)$ and $\phi_4(x)=2\sinh(\log(\log(1+e^x)))$, it proves convergence theorems showing root-exponential decay can be achieved with potentially larger domain widths $d$ and better decay rates $\mu$. Numerical experiments on representative test functions demonstrate faster convergence and smoother error behavior compared to the original Stenger formulas. The work advances high-order derivative approximation on unbounded domains, with prospects for further acceleration via double-exponential transformations in future research.

Abstract

F. Stenger proposed efficient approximation formulas for derivatives over infinite intervals. These formulas were derived by combining the Sinc approximation with appropriate conformal maps. It has been demonstrated that these formulas can attain root-exponential convergence. In this study, we enhance the convergence rate by improving the conformal maps employed in those formulas. We provide a theoretical error analysis and numerical experiments that confirm the effectiveness of our new formulas.

Improvement of conformal maps combined with the Sinc approximation for derivatives over infinite intervals

TL;DR

The paper targets efficient numerical differentiation over infinite intervals by enhancing Stenger's Sinc-approximation-based formulas with improved conformal maps. By replacing the i=2 and i=4 maps with and , it proves convergence theorems showing root-exponential decay can be achieved with potentially larger domain widths and better decay rates . Numerical experiments on representative test functions demonstrate faster convergence and smoother error behavior compared to the original Stenger formulas. The work advances high-order derivative approximation on unbounded domains, with prospects for further acceleration via double-exponential transformations in future research.

Abstract

F. Stenger proposed efficient approximation formulas for derivatives over infinite intervals. These formulas were derived by combining the Sinc approximation with appropriate conformal maps. It has been demonstrated that these formulas can attain root-exponential convergence. In this study, we enhance the convergence rate by improving the conformal maps employed in those formulas. We provide a theoretical error analysis and numerical experiments that confirm the effectiveness of our new formulas.

Paper Structure

This paper contains 10 sections, 25 theorems, 74 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Summary of existing and new results
  3. Convergence theorem of Stenger's formula
  4. Convergence theorem of our formula
  5. Numerical examples
  6. Proofs
  7. Sketch of the proof
  8. Proof of Theorem \ref{['thm:our-2']}
  9. Proof of Theorem \ref{['thm:our-4']}
  10. Concluding remarks

Key Result

Theorem 2.1

Assume that $f$ is analytic in $\psi_2(\mathscr{D}_d)$ with $0<d<\piup/2$, and that there exist positive constants $K$, $\alpha$ and $\beta$ such that holds for all $z\in\psi_2(\mathscr{D}_d)$. Let $\mu = \min\{\alpha,\beta\}$, let $M$ and $N$ be defined as and let $h$ be defined as Then, there exists a constant $C$ independent of $n$ such that

Figures (2)

  • Figure 1: Approximation errors of $f(t)$ in \ref{['eq:example1']}. $M$ and $N$ are defined by \ref{['eq:Def-MN']} with respect to $n$.
  • Figure 4: Approximation errors of $f(t)$ in \ref{['eq:example2']}. $M$ and $N$ are defined by \ref{['eq:Def-MN']} with respect to $n$.

Theorems & Definitions (39)

  • Theorem 2.1: Stenger stenger93:_numer
  • Theorem 2.2: Stenger Stenger
  • Theorem 2.3
  • Theorem 2.4
  • Remark 3.1
  • definition 1
  • Theorem 4.1: Okayama and Tanaka okayama23:_error_sinc
  • Theorem 4.2
  • Lemma 4.3: Stenger stenger93:_numer
  • Lemma 4.4: Okayama et al. OkaShinKatsu
  • ...and 29 more