Speaker
Description
In 2006, V.I. Danchenko suggested the method of $h$-sums
for approximation problems and numerical analysis (by $ h-sum$ $of$ $order$ $n$ we mean an expression of the form
$H_n(z)=\sum_{k=1}^n \lambda_k h(\lambda_k z), \quad \lambda_k\in \mathbb{C},$
where $h(z)$ is a fixed function, analytic in the disk $D=\{z:|z|<1\}$, and $\lambda_1,\dots,\lambda_n$ are independent numeric parameters).
In particular, it was proved that for every $n=1,2,\dots$ there are numbers $\lambda_{n,1},\dots,\lambda_{n,n}$ such that the corresponding $h$-sum $H_n$ interpolates the differentiation operator $(zh(z))'$ at the node $z_0=0$ with the multiplicity $n$:
$(zh(z))'=\sum_{k=1}^n\lambda_{n,k} h(\lambda_{n,k}z)+O(z^n),$
and $H_n(z)$ converges uniformly to $(zh(z))'$ as $n\to \infty$ on compact subsets $K\subset D$ with exponential rate.
We consider an analogous problem of numerical differentiation $ by$ $the$ $difference$ $of$ $two$ $h-sums$ $of$ $order$ $n$. It can be shown that under this modified approach the rate of approximation is much higher.
