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DE-Sinc approximation for unilateral rapidly decreasing functions and its computational error bound

Tomoaki Okayama

TL;DR

The paper addresses the efficiency of Sinc approximations for unilateral rapidly decreasing functions by introducing a double-exponential transform $t = \phi_5(x) = 2\sinh(\log(\log(1+e^{\pi\sinh x})))$. It proves an almost exponential convergence rate, $O(e^{-c n/\log n})$, with a computable error bound, under analytic and growth conditions on the target function. The main contribution is a rigorous error bound with explicit constants for the DE-Sinc formula, extending the existing Sinc framework and outperforming root-exponential rates. Numerical experiments on two unilateral decreasing functions confirm the theoretical results and demonstrate improved convergence and reliable error control. This work provides practically usable guarantees for high-accuracy Sinc approximations in unbounded unilateral settings and informs numerical integration and approximation tasks that rely on guaranteed accuracy.

Abstract

The Sinc approximation is known to be a highly efficient approximation formula for rapidly decreasing functions. For unilateral rapidly decreasing functions, which rapidly decrease as $x\to\infty$ but does not as $x\to-\infty$, an appropriate variable transformation makes the functions rapidly decreasing. As such a variable transformation, Stenger proposed $t = \sinh(\log(\operatorname{arsinh}(\exp x)))$, which enables the Sinc approximation to achieve root-exponential convergence. Recently, another variable transformation $t = 2\sinh(\log(\log(1+\exp x)))$ was proposed, which improved the convergence rate. Furthermore, its computational error bound was provided. However, this improvement was not significant because the convergence rate remained root-exponential. To improve the convergence rate significantly, this study proposes a new transformation, $t = 2\sinh(\log(\log(1+\exp(π\sinh x))))$, which is categorized as the double-exponential (DE) transformation. Furthermore, this study provides its computational error bound, which shows that the proposed approximation formula can achieve almost exponential convergence. Numerical experiments that confirm the theoretical result are also provided.

DE-Sinc approximation for unilateral rapidly decreasing functions and its computational error bound

TL;DR

The paper addresses the efficiency of Sinc approximations for unilateral rapidly decreasing functions by introducing a double-exponential transform . It proves an almost exponential convergence rate, , with a computable error bound, under analytic and growth conditions on the target function. The main contribution is a rigorous error bound with explicit constants for the DE-Sinc formula, extending the existing Sinc framework and outperforming root-exponential rates. Numerical experiments on two unilateral decreasing functions confirm the theoretical results and demonstrate improved convergence and reliable error control. This work provides practically usable guarantees for high-accuracy Sinc approximations in unbounded unilateral settings and informs numerical integration and approximation tasks that rely on guaranteed accuracy.

Abstract

The Sinc approximation is known to be a highly efficient approximation formula for rapidly decreasing functions. For unilateral rapidly decreasing functions, which rapidly decrease as but does not as , an appropriate variable transformation makes the functions rapidly decreasing. As such a variable transformation, Stenger proposed , which enables the Sinc approximation to achieve root-exponential convergence. Recently, another variable transformation was proposed, which improved the convergence rate. Furthermore, its computational error bound was provided. However, this improvement was not significant because the convergence rate remained root-exponential. To improve the convergence rate significantly, this study proposes a new transformation, , which is categorized as the double-exponential (DE) transformation. Furthermore, this study provides its computational error bound, which shows that the proposed approximation formula can achieve almost exponential convergence. Numerical experiments that confirm the theoretical result are also provided.

Paper Structure

This paper contains 12 sections, 21 theorems, 112 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and summary
  2. Summary of existing and new theorems
  3. Existing results
  4. New results
  5. Numerical examples
  6. Proofs
  7. Sketch of the proof
  8. Useful inequality on $\mathscr{D}_d^{+}$
  9. Useful inequalities on $\mathscr{D}_d^{-}$
  10. Useful inequalities on $\mathscr{D}_d$
  11. Bound of the discretization error (proof of Lemma \ref{['lem:bound-None']})
  12. Bound of the truncation error (proof of Lemma \ref{['lem:truncation-error']})

Key Result

Theorem 2.1

Assume that $f$ is analytic in $\psi_5(\mathscr{D}_d)$ with $0<d\leq\piup/2$, and that there exist positive constants $K_{+}$, $K_{-}$, $\alpha$ and $\beta$ such that holds for all $z\in\psi_5(\mathscr{D}_d^{+})$. Let $\mu = \min\{\alpha,\beta\}$, let $M$ and $N$ be defined by and let $h$ be defined by Then, there exists a constant $C$ independent of $n$ such that

Figures (1)

  • Figure 1: Observed errors and error bounds for $f_1(t)$ in \ref{['eq:f_1']}.

Theorems & Definitions (37)

  • Theorem 2.1
  • Theorem 2.2: Okayama and Shiraishi okayama21:_improv_sinc
  • Theorem 2.3
  • Remark 2.1
  • definition 1
  • Theorem 4.1: Stenger stenger93:_numer
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4: Okayama okayama22:_doubl
  • proof
  • ...and 27 more