Rate of approximation of $zf'(z)$ by special sums associated with the zeros of the Bessel polynomials

TL;DR

This paper introduces a new numerical differentiation formula using zeros of Bessel polynomials, achieving rapid convergence for analytic functions within the unit disk.

Contribution

The paper proposes a novel approximation formula for $zf'(z)$ based on Bessel polynomial zeros, with proven high convergence rate and exactness for polynomials up to degree 2n.

Findings

High rate of convergence $O(0.85^n n^{1-n})$ for the approximation

Exact for all polynomials of degree at most 2n

Effective for analytic functions within the unit disk

Abstract

Let $\alpha_{n1},\dots,\alpha_{nn}$ be the zeros of the $n$th Bessel polynomial $y_n(z)$ and let $a_{nk}=1-\alpha_{nk}/2$, $b_{nk}=1+\alpha_{nk}/2$ $(k=1,\dots,n)$. We propose the new formula \[z f'(z)\approx \sum_{k=1}^n \big(f(a_{nk} z)-f(b_{nk} z)\big)\] for numerical differentiation of analytic functions $f(z)=\sum_0^\infty f_m z^m$. This formula is exact for all polynomials of degree at most $2n$. We find the sharp order of nonlocal estimate of the corresponding remainder for the case when all $|f_m|\le 1$. The estimate shows a high rate of convergence of the differentiating sums to $zf'(z)$ on compact subsets of the open unit disk, namely, $O(0.85^n n^{1-n})$ as $n\to \infty$.