复平面上的傅立叶理论II：Weak Convergence，奇异性的分类和因式分解
Fourier Theory on the Complex Plane II: Weak Convergence, Classification and Factorization of Singularities
摘要：
利用内解析函数的泰勒级数讨论了既不是强收敛也不是强发散的DP傅里叶级数的收敛性.这些是内解析函数的泰勒级数的最大收敛圆盘是开单位圆盘的情况。根据相应的内解析函数的奇异性结构，建立了一大类DP傅里叶级数的收敛模式的基本完整的分类.给定一个DP实函数的弱收敛傅里叶级数，它示出了如何从它生成其他表达式，涉及三角级数，收敛到同一个功能，但具有更好的收敛特性。这是通过分解出相应的内部解析函数的奇异性的过程来完成的，并且甚至适用于发散的傅里叶级数。这可以被解释为一种重申技术，它被底层的分析结构牢牢地锚定。
英文摘要：
The convergence of DP Fourier series which are neither strongly convergent nor strongly divergent is discussed in terms of the Taylor series of the corresponding inner analytic functions. These are the cases in which the maximum disk of convergence of the Taylor series of the inner analytic function is the open unit disk. An essentially complete classification, in terms of the singularity structure of the corresponding inner analytic functions, of the modes of convergence of a large class of DP Fourier series, is established. Given a weakly convergent Fourier series of a DP real function, it is shown how to generate from it other expressions involving trigonometric series, that converge to that same function, but with much better convergence characteristics. This is done by a procedure of factoring out the singularities of the corresponding inner analytic function, and works even for divergent Fourier series. This can be interpreted as a resummation technique, which is firmly anchored by the underlying analytic structure.
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