Convergence Problems and Integral Inequalities

该项目有两个目的：第一个问题是确定所谓Zygmund类的Fourier或沃尔什Fourier级数的收敛性，第二个问题是得到可用于各种数学分析场合的积分不等式，第一个问题是很难解决的，所以我们的主要工作围绕第二个问题展开，与数学分析的其他分支交织在一起，尤其是函数逼近理论。由于很少有结果的报告属于空间BMO的函数的逼近，首席研究员试图弥补讲座笔记的主题，这将被附在这份报告中，证明了伯恩斯坦不等式在这种设置，然后它是改进的，以便产生易于处理的应用。因此，整个理论是建立在几个简单的“工具”上的，而这将显示出伯恩斯坦思想的广阔范围。

The purpose of the project was twofold : first, to determine the convergence behavior of the Fourier or Walsh Fourier series of so-called Zygmund class and second, to obtain integral inequalities which can be used in various scenes of mathmatical analysis.The first problem turns out to be very difficult to attack, so our main effort centered around the second problem, intertwining with other branches of mathematical analysis, especially the theory of approximation of functions. Since very few results are reported on the approximation of functions belonging to the space BMO, the chief investigator tried to make up lecture notes on the topic, which will be appended in this report.The Bernstein inequality is proved in this setting, then it is improved so as to yield easy treatment for applications. As a result, the whole theory is based on a few simple "tools", and this will show the extent of the vast extent of Bernstein's ideas.