众所周知，调和分析一直是分析学的重要方向之一。随着调和分析理论的发展和逐步完善，它在偏微分方程、群论、数论等数学领域，以及信号处理、网络安全等工程领域中找到了广泛而深入的应用。

本篇论文首先介绍经典傅立叶分析的一些基本理论，傅立叶分析有两个基本问题，函数要满足什么条件，其傅立叶级数才收敛；如果一个三角级数收敛，如何判断它是否是某个函数的傅立叶级数。我们在第三章将对它们进行研究。

接下来本篇论文主要研究允许依范数收敛的齐性巴拿赫空间，举出一些特殊的齐性巴拿赫空间的例子，再判定它们是不是允许依范数收敛。因为傅立叶级数的收敛性，比相应的关于特殊的可积核的可积性要精确的多。而依范数收敛又比逐点收敛容易。

                                         

关键词 可积核；齐性巴拿赫空间；允许依范数收敛；允许共轭

 

 

As we all know, harmonic analysis has always been one of the important courses in analysis. With the development of the theory of harmonic analysis and its becoming complete, it have been widely and deeply used in partially differentiate, group theory, number theory and some mathematical fields, even in some industry fields like signal processing, network safety and so on.

In this paper, we will introduce some basic theories of classic Fourier analysis first, there are two basic problems on Fourier analysis, one is what properties of the function the Fourier series’ convergence follows from; the other is if a function is convergence, how to get the criterion for whether it is some function’s Fourier series or not. In chapter III, we will analysis them.

Then, this paper will focus on homogeneous Banach space admitting convergence in norm. Firstly, we take some examples of homogeneous Banach space, then show that whether them admit convergence in norm or not. Because the problems of convergence of Fourier series are far more delicate than the corresponding problems of summability with respect to “good” summability kernels, and the problems of convergence in norm are usually easier than those of pointwise convergence.

 

Key words　summability kernel  homogeneous Banach space  admit convergence in norm  admit conjugate

 

 

 

 

 

 

 

 

 

 

 

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