Improvement of selection formulas of mesh size and truncation numbers for the DE-Sinc approximation and its theoretical error bound

The Sinc approximation applied to double-exponentially decaying functions is referred to as the DE-Sinc approximation. This approximation has notably been utilized for many applications because of its high efficiency. The Sinc approximation's mesh size and truncation numbers should be optimally selected to avail its full performance. However, the usual formula has only been ``near-optimally'' selected because the optimal formula between the two cannot be expressed in terms of elementary functions. In this study, we propose two improved formulas. The first one is based on the concept by an earlier research that produced an improved selection formula for the double-exponential formula. The formula performed better than the usual one, but was still not optimal. As a second formula, we introduce a new parameter to propose a truly optimal formula between the two. We give explicit error bounds for both formulas. Numerical comparisons show that the first formula gives a better error bound than the standard formula, and the second formula gives a far better error bound than both the standard and first formulas.