Design of accurate formulas for approximating functions in weighted Hardy spaces by discrete energy minimization
For researchers in approximation theory and numerical analysis, this provides a practical method to design near-optimal sampling points for weighted Hardy spaces.
The paper proposes a method for designing approximation formulas for weighted analytic functions by minimizing a discrete energy related to the minimum worst error. The resulting formulas outperform sinc approximation in numerical experiments.
We propose a simple and effective method for designing approximation formulas for weighted analytic functions. We consider spaces of such functions according to weight functions expressing the decay properties of the functions. Then, we adopt the minimum worst error of the $n$-point approximation formulas in each space for characterizing the optimal sampling points for the approximation. In order to obtain approximately optimal sampling points, we consider minimization of a discrete energy related to the minimum worst error. Consequently, we obtain an approximation formula and its theoretical error estimate in each space. In addition, from some numerical experiments, we observe that the formula generated by the proposed method outperforms the corresponding formula derived with sinc approximation, which is near optimal in each space.