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

Abstract

The Sinc approximation is highly effective for functions that decay rapidly at both ends of the real axis. For unilateral rapidly decreasing functions, which decay algebraically as $t\to-\infty$ and exponentially as $t\to\infty$, an appropriate variable transformation is required. Existing single-exponential transformations for this class of functions yield only root-exponential convergence, even when improved transformations are employed. This paper develops a double-exponential (DE)-Sinc approximation based on the transformation $t=2\sinh(\log(\log(1+\exp(π\sinh x))))$, which was previously introduced for numerical integration. The main contribution is a rigorous and computable error bound of order $\operatorname{O}(\exp(-cn/\log n))$ with a constant explicitly expressed in terms of the problem parameters. Under the stated assumptions, the resulting approximation achieves almost exponential convergence and is suitable for computation with guaranteed accuracy. Numerical examples satisfying these assumptions confirm the predicted convergence behavior and the validity of the derived error bound.