Approximations of zeros of entire functions by zeros of polynomials
Let \[ f(\lambda):=\sum_{k=0}^{\infty}\frac{a_k\lambda^k}{(k!)^{\rho}}\quad (\lambda \in \mathbb C, a_0=1, \rho >1/2) \] denote an entire function. Set \[ p_n(\lambda):=\sum_{k=0}^{n}\frac{a_k\lambda^k}{(k!)^{\rho}}\quad \text{and}\quad q_n:=\left (\sum_{k=n+1}^{\infty}|a_k|^2\right)^{1/2}. \] In the paper under review, the author proposes a new approach to the following question. If \(q_n\) is small, how close are the zeros of \(p_n\) to the zeros of \(f\)? The author's approach is based on recent estimates for the norm of the resolvent of a Hilbert--Schmidt operator.
scientific article; zbMATH DE number 1538245