本文關(guān)鍵詞： 四元數(shù) Slice正則函數(shù) Bieberbach猜測(cè) 增長(zhǎng)定理 Bloch-Landau定理 Bernstein不等式 Schwarz引理 Bloch空間 測(cè)不準(zhǔn)原理　出處：《中國(guó)科學(xué)技術(shù)大學(xué)》2017年博士論文　論文類型：學(xué)位論文

【摘要】：本文主要研究復(fù)分析在高維非交換代數(shù)上的推廣,其中包括以下三個(gè)方面:(1)slice正則函數(shù)的幾何函數(shù)論;(2)slice正則函數(shù)的函數(shù)空間論;(3)四元數(shù)Hilbert空間中的測(cè)不準(zhǔn)原理.全文共分為五章.第一章是緒論,介紹了本論文的研究背景和所取得的成果.第二章給出了本論文中常用的符號(hào)、概念和結(jié)論.第三章主要研究了 slice正則函數(shù)的幾何函數(shù)論.本章首先在四元數(shù)slice正則函數(shù)中定義了 slice星形函數(shù),slice近凸函數(shù),slice螺形函數(shù),證明了 Bieberbach猜測(cè)對(duì)slice近凸函數(shù)是成立的,對(duì)slice星形函數(shù)建立了 Fekete-Szego不等式、增長(zhǎng)定理、掩蓋定理和偏差定理.其次,本章研究了.類交錯(cuò)代數(shù)上slice正則函數(shù)的增長(zhǎng)定理和偏差定理.然后,針對(duì)四元數(shù)slice正則函數(shù)建立了三類Bloch-Landau型定理并推廣了經(jīng)典的Bernstein不等式.最后,本章圍繞Schwarz引理在高維中的推廣.特別地,研究了 slice Clifford分析以及多次調(diào)和函數(shù)中的Schwarz引理及其邊界行為.第四章研究了 α-Bloch函數(shù)在高維空間中的兩類推廣.一方面,研究了無(wú)限維Hilbert空間單位球上的全純?chǔ)?Bloch函數(shù),定義了四種范數(shù)并證明了其等價(jià)性.作為應(yīng)用,建立了無(wú)限維Hilbert空間中的Hardy-Littlewood定理.另一方面,研究了四元數(shù)單位球上的正則α-Bloch函數(shù),建立了相應(yīng)的Forelli-Rudin估計(jì),Hardy-Littlewood定理,并對(duì)其對(duì)偶空間進(jìn)行了研究.第五章建立了四元數(shù)Hilbert空間中的測(cè)不準(zhǔn)原理.
[Abstract]:In this paper, we mainly study the generalization of complex analysis on high dimensional noncommutative algebras. It includes the following three aspects: geometric function theory of slice regular function and function space theory of 2slice regular function. The uncertainty principle in Hilbert space of quaternion is discussed. The whole paper is divided into five chapters. Chapter one is an introduction. The research background and achievements of this thesis are introduced. In chapter 2, the commonly used symbols in this paper are given. In chapter 3, the geometric function theory of slice regular function is studied. In this chapter, the slice star function is defined in the quaternion slice regular function. It is proved that Bieberbach conjecture is true for slice near-convex functions, and Fekete-Szego inequality, growth theorem, concealment theorem and deviation theorem are established for slice star functions. In this chapter, we study the growth theorems and deviation theorems of slice regular functions on quasi-staggered algebras. Then, we establish three Bloch-Landau type theorems for quaternion slice regular functions and generalize the classical Bernstein inequalities. This chapter focuses on the generalization of Schwarz Lemma in higher dimensions. In particular, we study the Schwarz Lemma and its boundary behavior in slice Clifford analysis and multiharmonic functions. In Chapter 4th, we study two generalizations of 偽 -Bloch functions in high dimensional spaces. In this paper, the holomorphic 偽 -Bloch function on the unit sphere of infinite dimensional Hilbert space is studied, four kinds of norms are defined and its equivalence is proved. As an application, the Hardy-Littlewood theorem in infinite dimensional Hilbert space is established. In this paper, we study the regular 偽 -Bloch function on the unit sphere of quaternions, establish the corresponding Forelli-Rudin estimate Hardy-Littlewood theorem, and study its dual space. Chapter 5th establishes the uncertainty principle in the quaternion Hilbert space.
【學(xué)位授予單位】：中國(guó)科學(xué)技術(shù)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O174.5

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本文編號(hào)：1533687