Infinite Order Differential Operators with a Glimpse to Applications to Superoscillations
Infinite Order Differential Operators with a Glimpse to Applications to Superoscillations
复制标题
无限阶微分算子及其在超级振荡中的应用概览
DOI：
10.1007/978-3-031-21460-8_1
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Struppa Daniele C.
中科院分区：
文献类型：
--
作者：
Aoki Takashi;Okada Yasunori;Sabadini Irene;Struppa Daniele C.
In this paper we will consider operators that can be formally written as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \begin{aligned} P\left(z,\frac{d}{dz}\right):=\sum_{n=0}^{\infty}a_n(z)\frac{d^n}{dz^n} \end{aligned}$$\end{document} where the functionsanare entire functions on the complex plane (possibly satisfying suitable growth conditions), and we will study their action on suitable spaces of entire functions as well. After an introductory section, whose content is well known, that describes infinite order differential operators from the point of view of the theory of hyperfunctions, we will describe as well convolutors arising from analytic functionals, and how they can be represented by infinite series of derivatives. The next section is dedicated to the way in which the study of longevity phenomena for superoscillations has led to a renewed interest for the theory of infinite order differential operators, and will present some recent results on the continuity of such operators. Finally, we will show how the idea of infinite order differential operators extends fruitfully to the hypercomplex setting.
In this paper we will consider operators that can be formally written as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \begin{aligned} P\left(z,\frac{d}{dz}\right):=\sum_{n=0}^{\infty}a_n(z)\frac{d^n}{dz^n} \end{aligned}$$\end{document} where the functionsanare entire functions on the complex plane (possibly satisfying suitable growth conditions), and we will study their action on suitable spaces of entire functions as well. After an introductory section, whose content is well known, that describes infinite order differential operators from the point of view of the theory of hyperfunctions, we will describe as well convolutors arising from analytic functionals, and how they can be represented by infinite series of derivatives. The next section is dedicated to the way in which the study of longevity phenomena for superoscillations has led to a renewed interest for the theory of infinite order differential operators, and will present some recent results on the continuity of such operators. Finally, we will show how the idea of infinite order differential operators extends fruitfully to the hypercomplex setting.