arXiv subjects

Min-Hee Kim

Publications and source records attributed to Min-Hee Kim.

2 recordsLinked to original sources

arXiv2019

The convergence of a sequence of polynomials and the distribution of their zeros

Suppose that $\langle f_n \rangle$ is a sequence of polynomials, $\langle f_n^{(k)}(0)\rangle$ converges for every non-negative integer $k$, and that the limit is not $0$ for some $k$. It is shown that if all the zeros of $f_1, f_2, \dots$ lie in the closed upper half plane $\rm{Im}\ z\geq 0$, or if $f_1, f_2, \dots$ are real polynomials and the numbers of their non-real zeros are uniformly bounded, then the sequence converges uniformly on compact sets in the complex plane. The results imply a theorem of Benz and a conjecture of Pólya.

math.CV↗

arXiv2015

On the Pólya-Wiman properties of Differential Operators

Let $ϕ(x)=\sum α_n x^n$ be a formal power series with real coefficients, and let $D$ denote differentiation. It is shown that "for every real polynomial $f$ there is a positive integer $m_0$ such that $ϕ(D)^mf$ has only real zeros whenever $m\geq m_0$" if and only if "$α_0=0$ or $2α_0α_2 - α_1^2 <0$", and that if $ϕ$ does not represent a Laguerre-Pólya function, then there is a Laguerre-Pólya function $f$ of genus $0$ such that for every positive integer $m$, $ϕ(D)^mf$ represents a real entire function having infnitely many nonreal zeros.

math.CV↗