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On a sufficient condition for a Mittag-Leffler function to have real zeros only, and the P\'olya-Wiman properties of differential operators : 주어진 미탁-레플러 함수가 실근만을 갖기 위한 충분조건과 미분 연산자의 폴랴-위만 성질
DC Field | Value | Language |
---|---|---|
dc.contributor.advisor | 김영원 | - |
dc.contributor.author | 김민희 | - |
dc.date.accessioned | 2017-07-14T00:42:00Z | - |
dc.date.available | 2017-07-14T00:42:00Z | - |
dc.date.issued | 2016-02 | - |
dc.identifier.other | 000000132316 | - |
dc.identifier.uri | https://hdl.handle.net/10371/121302 | - |
dc.description | 학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2016. 2. 김영원. | - |
dc.description.abstract | In this dissertation, we study the distribution of zeros of entire functions.
First, we study the reality of zeros of Mittag-Leffler functions. If $\alpha$ and $\beta$ are complex numbers with $\mathrm{Re}\ \alpha>0$, the Mittag-Leffler function $E_{\alpha,\beta}$ is defined by \begin{equation*} E_{\alpha,\beta} (z)=\sum_{k=0}^{\infty} \frac{z^k}{\Gamma{(\beta+\alpha k })}. \end{equation*} One of the most recent results on the zeros of the Mittag-Leffler functions is due to Popov and Sedletskii: if $\alpha > 2$ and $0 < \beta \leq 2\alpha - 1$ or if $\alpha > 4$ and $0 < \beta \leq 2\alpha$ then $ E_{\alpha,\beta}(z)$ has only real zeros. We improve the result by showing that if $\alpha\geq4.07$ and $0<\beta\leq 3\alpha$ then $E_{\alpha,\beta}(z)$ has only real zeros. %We also obtain some asymptotic results on the distribution of zeros of Mittag-Leffler functions in %the general case where is $\alpha >2$. Second, we study the P\'olya-Wiman properties of differential operators. Let $\phi(x)=\sum \alpha_n x^n$ 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 $\phi(D)^mf$ has only real zeros whenever $m\geq m_0$'' if and only if ``$\alpha_0=0$ or $2\alpha_0\alpha_2 - \alpha_1^2 <0$'', and that if $\phi$ does not represent a Laguerre-P\'olya function, then there is a Laguerre-P\'olya function $f$ of genus $0$ such that for every positive integer $m$, $\phi(D)^mf$ represents a real entire function having infinitely many nonreal zeros. Finally, we prove the identity $$ \sup\{\alpha\in\mathbb{R}:e^{\alpha D^2}\cos{ D}\ M^n \mbox{\ has real zeros only}\}=4{\lambda_n}^{-2}, $$ where $M^n $ is the monic monomial of degree $n$, that is, $M^n (z)=z^n$, and $\lambda_n$ is the largest zero of the $2n$-th Hermite polynomial $H_{2n}$ given by $$ H_{2n}(z)=(2n)!\sum_{k=0}^{n}\frac{(-1)^k}{k!(2n-2k)!}(2z)^{2n-2k}. $$ | - |
dc.description.tableofcontents | Introduction 1
Chapter 1 Sufficient condition for a Mittag-Leffler function to have real zeros only 7 1.1 Main result and sketch outline of the proof 7 1.2 Sufficient condition to have real zeros only 11 1.3 Proof of Theorem 1.1.4 in the case $n<\lfloor\alpha/4\rfloor$ 19 1.3.1 Proof of Proposition 1.3.8 29 1.4 Proof of Theorem 1.1.5 37 Chapter 2 P\'olya-Wiman properties of differential operators 40 2.1 P\'olya-Wiman property 40 2.2 P\'olya-Wiman property with respect to real polynomials 43 2.2.1 Proof of Theorem 2.2.3 46 2.2.2 Proof of Theorem 2.2.4 48 2.2.3 Laguerre-P\'olya class and P\'olya-Wiman property with respect to real polynomials 50 2.3 P\'olya-Wiman property with respect to Laguerre -P\'olya functions of genus $0$ 54 2.4 Asymptotic behavior of distribution of zeros of $\phi (D)^m f $ as $m\to \infty$ 63 Chapter 3 Asymptotic behavior of distribution of the zeros of a one-parameter family of polynomials 65 3.1 Asymptotic behavior of distribution of the zeros of $\phi (D)^m f $ as $m\to \infty $ 65 3.2 Zeros of polynomials with complex coefficients 66 3.3 Proofs of Theorem 3.1.1 and Theorem 3.1.2 68 Chapter 4 De Bruijn-Newman constant of the polynomial $(z+i)^n +(z-i)^n$ 75 4.1 Main Result 75 4.2 Preliminaries 77 4.3 Proof of the main result 78 Bibliography 82 Abstract (in Korean) 85 | - |
dc.format | application/pdf | - |
dc.format.extent | 3116711 bytes | - |
dc.format.medium | application/pdf | - |
dc.language.iso | en | - |
dc.publisher | 서울대학교 대학원 | - |
dc.subject | Mittag-Leffler functions | - |
dc.subject | P\'olya-Wiman Theorem | - |
dc.subject | zeros of polynomials and entire functions | - |
dc.subject | linear differential operators | - |
dc.subject | Laguerre-P\'olya class | - |
dc.subject | Hermite polynomials | - |
dc.subject | De Bruijn-Newman constant | - |
dc.subject.ddc | 510 | - |
dc.title | On a sufficient condition for a Mittag-Leffler function to have real zeros only, and the P\'olya-Wiman properties of differential operators | - |
dc.title.alternative | 주어진 미탁-레플러 함수가 실근만을 갖기 위한 충분조건과 미분 연산자의 폴랴-위만 성질 | - |
dc.type | Thesis | - |
dc.contributor.AlternativeAuthor | Min-Hee Kim | - |
dc.description.degree | Doctor | - |
dc.citation.pages | 84 | - |
dc.contributor.affiliation | 자연과학대학 수리과학부 | - |
dc.date.awarded | 2016-02 | - |
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