Rate of approximation of $z{f}^{\prime }(z)$ by special sums associated with the zeros of the Bessel polynomials

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 $z{f}^{\prime }(z)$ on compact subsets of the open unit disk, namely, $O(0.85^n{n}^{1-n})$ as $n\to \infty$.