论文摘要
本文首先研究了带无穷远点一般增长性条件的正实轴上的Riemann-Hilbert边值问题.为研究该问题我们给出了C[0,+∞)上的解析函数在无穷远点及原点主部和阶的定义,讨论了正实轴上Cauchy型积分在无穷远点和原点的性质以及它在正实轴上正负边值的性质.在此基础上给出了带无穷远点一般增长性条件的正实轴上Riemann-Hilbert边值问题的合理提法并进行了详细地求解.其次,我们介绍了矩阵值Riemann-Hilbert边值问题,讨论了三种特殊的矩阵值Riemann-Hilbert边值问题,尤其是正实轴上下三角矩阵值Riemann-Hilbert边值问题,得到正实轴上关于某类权函数正交的多项式的特征刻划.第三,我们证明了带变化大负参数广义Bessel多项式在正实轴上的正交性,从而得到此多项式的特征刻划.第四,借助于一些辅助函数及抛物柱面函数,构造拟基本解,进而得出带变化大负参数广义Bessel多项式的渐近展开式.全文共分为六章.第一章介绍研究背景.第二章介绍预备知识,包括广义Bessel多项式简介、解析函数边值理论简介、几类特殊函数简介等.第三章研究带无穷远点增长性条件的正实轴上的Riemann-Hilbert边值问题.第四章介绍了矩阵值Riemann-Hilbert边值问题,特别讨论了正实轴上下三角矩阵值Riemann-Hilbert边值问题.第五章证明了带变化大负参数广义Bessel多项式在正实轴上的正交性,得到该多项式的特征刻划.第六章引入辅助函数,构造拟基本解,得出带变化大负参数广义Bessel多项式的渐近展开式.
论文目录
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