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

This paper introduces a double-exponential Sinc approximation tailored for unilateral rapidly decreasing functions, establishing a rigorous, computable error bound that guarantees almost exponential convergence of order $O(\exp(-cn/\log n))$ and validating the method through numerical examples.

Original authors: Tomoaki Okayama

Published 2026-07-28

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Tomoaki Okayama

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a faint, whispering song that plays across an infinite stage. The song is loud and clear in the middle, but as you move toward the far left and right edges of the stage, the volume drops off. In the world of mathematics and computer science, this "song" is a function, and the "stage" is the number line. Scientists often need to recreate these songs perfectly using a limited set of notes (data points) to solve complex problems, like predicting how heat spreads or how a bridge vibrates. The challenge is that if the song fades away too slowly on one side, or too quickly on the other, standard tools get confused and make mistakes. This is the realm of numerical analysis, where mathematicians build digital microscopes to see the invisible details of these functions. The goal is always the same: get the most accurate picture possible using the fewest number of notes, because every extra note costs time and computing power.

For decades, a clever tool called the "Sinc approximation" has been the go-to method for listening to these fading songs. Think of the Sinc method as a master tailor who can stitch together a perfect fabric from a few scattered threads. However, this tailor has a specific weakness: if the song fades away in a weird, lopsided way—dropping off slowly on the left but vanishing instantly on the right—the standard tailor's pattern fails to fit. Previous attempts to fix this involved using a "single-exponential" transformation, which is like stretching the stage so the fading song fits better. But even with this stretch, the tailor could only achieve a "root-exponential" speed. In plain English, this means the accuracy improves, but it's like climbing a steep hill where every step gets you only a tiny bit closer to the top. You need a massive number of steps (computing power) to get a truly precise result.

This paper introduces a new, super-powered tailor and a completely new way to stretch the stage. The author, Tomoaki Okayama, proposes a "Double-Exponential" (DE) transformation. If the old method was a standard bicycle, this new method is a rocket ship. Instead of just stretching the stage, this new transformation warps the space so drastically that the "fading" parts of the song are compressed into a tiny, manageable corner. The result is a mathematical breakthrough for a specific class of "unilateral rapidly decreasing functions" (those lopsided songs). The paper proves that with this new method, the error doesn't just shrink; it vanishes at an "almost exponential" rate. This means you can get the same high-precision accuracy with significantly fewer data points than before.

The paper doesn't just claim this works; it provides a rigorous, mathematically ironclad proof. The authors derived a specific "error bound"—a guarantee that says, "If you use this method, your mistake will never be larger than this specific number." This is crucial because in science, knowing how wrong you might be is just as important as knowing the answer. The paper explicitly rules out the idea that the old "single-exponential" methods are sufficient for high-precision needs in these specific cases, showing they are inherently slower. While the paper confirms this speed-up through computer simulations with three specific test functions, it also notes a boundary: if a function is too weird (specifically, if it doesn't meet certain smoothness rules in the complex plane), this new rocket ship might not fly as predicted, and the error guarantee might not hold. However, for the vast majority of functions that fit the description, the paper demonstrates that this new DE-Sinc approximation is a game-changer, offering a path to "computation with guaranteed accuracy" that is both faster and more reliable than anything previously available for these tricky, one-sided problems.

Technical Summary: DE-Sinc Approximation for Unilateral Rapidly Decreasing Functions

Problem Statement
The Sinc approximation method is highly effective for functions that decay rapidly at both ends of the real axis. However, for unilateral rapidly decreasing functions—defined as functions that decay algebraically as $t \to -\infty$ and exponentially as $t \to \infty$ —standard variable transformations yield only root-exponential convergence of order $O(\exp(-c\sqrt{n}))$ . While improved single-exponential (SE) transformations exist for this class of functions, they do not overcome this asymptotic limitation. Furthermore, while Double-Exponential (DE) transformations have been successfully applied to other cases (finite intervals, symmetric decay, etc.) to achieve almost exponential convergence, a rigorous, computable error bound for the DE-Sinc approximation of unilateral rapidly decreasing functions (Case 5 in Stenger's classification) had not been established.

Methodology
This paper proposes a DE-Sinc approximation based on a specific double-exponential variable transformation:
$t = \phi_5(x) = 2 \sinh(\log(\log(1 + e^{\pi \sinh x})))$
This transformation was previously introduced for numerical integration but is here adapted for function approximation. The methodology involves:

Transformation: Mapping the original function $f(t)$ to a transformed function $F(x) = f(\phi_5(x))$ defined on the real axis.
Approximation: Applying the standard Sinc series expansion to $F(x)$ and mapping the result back to the $t$ -domain.
Error Analysis: Decomposing the total error into discretization error and truncation error.
- The discretization error is bounded using the norm of $F$ in the Hardy space $H_1(D_d)$ over a strip domain $D_d$ .
- The truncation error is bounded by analyzing the decay properties of $F(x)$ at $\pm \infty$ .
Technical Challenges: The proof requires new estimates due to the nested logarithmic structure of $\phi_5$ , its branch structure in the complex plane, and the asymmetric decay behavior at the two ends of the real axis. The authors derive explicit inequalities for the transformed function's behavior on the strip domain to bound the constants rigorously.

Key Contributions

New Theorem (Theorem 2.3): The paper establishes a rigorous error bound for the DE-Sinc approximation of unilateral rapidly decreasing functions. The bound is of the form:
$\sup_{t \in \mathbb{R}} |f(t) - \text{Approx}(t)| \leq C e^{-\pi d n / \log(2dn/\mu)}$
where $n$ is the number of sampling points, $d$ is the strip width, and $\mu$ relates to the decay rates.
Explicit Constants: Unlike previous theoretical results that may rely on non-computable constants, this paper provides a formula for the constant $C$ explicitly in terms of the problem parameters ( $K_+, K_-, \alpha, \beta, d$ ). This makes the error bound computable and suitable for algorithms requiring guaranteed accuracy.
Convergence Rate: The method achieves almost exponential convergence of order $O(\exp(-cn/\log n))$ . This is a significant improvement over the root-exponential convergence $O(\exp(-c\sqrt{n}))$ of existing SE-Sinc methods for this class of functions.
Sample Complexity: To achieve an accuracy $\epsilon$ , the proposed DE-Sinc method requires $n = O(\log(1/\epsilon) \log \log(1/\epsilon))$ sampling points, compared to $n = O((\log(1/\epsilon))^2)$ for the SE-Sinc approximation.

Results
The paper validates the theoretical findings through numerical examples using three specific unilateral rapidly decreasing functions ( $f_1, f_2, f_3$ ):

For functions $f_1$ and $f_2$ , which satisfy the analyticity assumptions of the new theorem, the observed errors converge significantly faster with the DE-Sinc approximation (Theorem 2.3) than with SE-Sinc approximations (Theorems 2.1 and 2.2).
The observed errors for these functions fall within the derived theoretical error bounds, confirming the validity of the constant $C$ and the convergence order.
For function $f_3$ , which does not satisfy the strict analyticity assumptions of the new theorem, the DE-Sinc approximation still performed competitively, though it did not exhibit the characteristic almost-exponential convergence, and the theoretical error bound was not applicable.

Significance
The paper claims that this work fills a critical gap in Sinc numerical methods by providing the first computable error bound for the DE-Sinc approximation of unilateral rapidly decreasing functions. The primary significance lies in enabling computation with guaranteed accuracy for this specific class of problems. By establishing almost exponential convergence, the method substantially reduces the computational cost (number of sampling points) required to achieve high precision compared to existing single-exponential approaches. The authors note that since Sinc approximation is a foundational component of quadrature and collocation methods, these results may serve as an analytical foundation for future guaranteed-accuracy algorithms for differential and integral equations involving unilateral decay.

Technical Summary: DE-Sinc Approximation for Unilateral Rapidly Decreasing Functions

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