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

1. Problem Statement

The paper addresses the numerical differentiation of functions defined over infinite intervals, specifically $(0, \infty)$ and $(-\infty, \infty)$. The standard approach involves the Sinc approximation, which combines a Sinc function basis with a conformal map to transform the infinite domain into a finite or more manageable one.

F. Stenger previously established efficient approximation formulas for derivatives by combining Sinc approximation with specific conformal maps ($\psi_i$). These formulas achieve root-exponential convergence of the order $O(\exp(-c\sqrt{n}))$, where $n$ is the number of discretization points. However, the convergence rate depends heavily on the parameters $d$ (the width of the analyticity strip in the complex plane) and $\mu$ (related to the decay rate of the function).

The specific problem targeted in this study is the suboptimal convergence rate in two specific cases identified by Stenger:

• Case 2: Interval $(0, \infty)$ where the function decays exponentially as $t \to \infty$. Stenger's map: $\psi_2(x) = \text{arsinh}(e^x)$.
• Case 4: Interval $(-\infty, \infty)$ where the function decays algebraically as $t \to -\infty$ and exponentially as $t \to \infty$. Stenger's map: $\psi_4(x) = \sinh(\log(\text{arsinh}(e^x)))$.

In these cases, the existing conformal maps limit the achievable values of $d$ and $\mu$, thereby capping the convergence speed.

2. Methodology

The authors propose a novel approach by replacing Stenger's conformal maps with new, improved maps while retaining the Sinc approximation framework.

Proposed Conformal Maps

• For Case 2 ($t \in (0, \infty)$):
  Replace $\psi_2(x) = \text{arsinh}(e^x)$ with:
  $$\phi_2(x) = \log(1 + e^x)$$
• For Case 4 ($t \in (-\infty, \infty)$):
  Replace $\psi_4(x) = \sinh(\log(\text{arsinh}(e^x)))$ with:
  $$\phi_4(x) = 2 \sinh(\log(\log(1 + e^x)))$$

Theoretical Framework

The authors derive new approximation formulas for the $m$-th derivative $f^{(m)}(t)$:
$$f^{(m)}(t) \approx \sum_{k=-M}^{N} \frac{f(\phi_i(kh))}{g_i(\phi_i(kh))} \left( \frac{d}{dt} \right)^m \left\{ g_i(t) S(k, h)(\phi_i^{-1}(t)) \right\}$$
where $g_i(t)$ are weight functions chosen to suppress divergence at endpoints, and $S(k, h)$ is the Sinc function.

The core of the methodology involves a rigorous theoretical error analysis to prove that these new maps allow for larger values of the parameters $d$ and $\mu$ in the convergence bound. The proof strategy involves:

1. Discretization Error: Utilizing the function space $H_1(D_d)$ and Faà di Bruno's formula to bound the derivatives of the composite functions involving the new inverse maps.
2. Truncation Error: Analyzing the decay of the function $F(\zeta) = f(\phi(\zeta))/g(\phi(\zeta))$ in the complex strip to establish bounds on the tail of the series.
3. Synthesis: Combining these bounds to derive the final error estimate.

3. Key Contributions

1. New Conformal Maps: The introduction of $\phi_2$ and $\phi_4$ as superior alternatives to Stenger's $\psi_2$ and $\psi_4$ for derivative approximation.
2. Enhanced Convergence Theorems:
   • Theorem 3.1 (Case 2): Proves convergence with $d < \pi$ (compared to Stenger's $d < \pi/2$).
   • Theorem 3.2 (Case 4): Proves convergence with $d < \pi$ and allows for a larger $\mu$ (specifically, the decay parameter $\beta$ can be effectively doubled compared to Stenger's formulation due to the structure of the new map).
   • The error bound remains $O(n^{(m+1)/2} \exp(-\sqrt{\pi d \mu n}))$, but the increased $d$ and $\mu$ result in a significantly smaller constant in the exponent, accelerating convergence.
3. Rigorous Proofs: Detailed proofs using complex analysis, specifically handling the singularities and decay properties of the new inverse maps $\phi_i^{-1}(t)$ and their derivatives.

4. Results

The authors conducted numerical experiments using double-precision arithmetic (and 20-digit precision for specific derivative calculations to avoid noise) to validate the theory.

• Test Case 1 (Case 2): $f(t) = \sqrt{\frac{t}{1+t}} e^{-t}(1-e^{-t})^2$ on $(0, \infty)$.
  • Parameters: Stenger ($d=1.57, \mu=0.5$) vs. Improved ($d=3.14, \mu=0.5$).
  • Outcome: The improved formula converged significantly faster. The error decreased more rapidly as the number of function evaluations ($M+N+1$) increased.
• Test Case 2 (Case 4): $f(t) = \frac{1}{(4+t^2)(1+e^{\pi t/2})}$ on $(-\infty, \infty)$.
  • Parameters: Stenger ($d=1.57, \mu \approx 0.785$) vs. Improved ($d=2.07, \mu \approx 1.57$).
  • Outcome: The improved formula demonstrated faster convergence. Additionally, the error profile of Stenger's formula appeared "bumpy" (less smooth), whereas the improved formula showed a smooth, monotonic convergence profile.

In both cases, the improved formula (1.6) consistently outperformed Stenger's formula (1.5), confirming the theoretical prediction that larger $d$ and $\mu$ values lead to superior performance.

5. Significance

• Efficiency: The proposed method achieves the same level of accuracy with fewer function evaluations compared to Stenger's original method. This is crucial for solving differential equations where high-order derivatives are required repeatedly.
• Theoretical Advancement: The paper demonstrates that the choice of conformal map is critical even within the "single-exponential" transformation class. It provides a blueprint for optimizing these maps by analyzing the analyticity strip width ($d$) and decay rates ($\mu$).
• Future Directions: The authors note that while single-exponential transformations have been improved, further gains might be possible by switching to double-exponential transformations (which are known to accelerate convergence in integral approximations). They indicate this is the subject of their ongoing research.

In summary, this work refines a well-established numerical technique (Sinc approximation for derivatives) by optimizing the underlying coordinate transformation, resulting in provably faster convergence rates for functions on infinite domains.