摘要
本文由以下三个部分组成:
第一部分:分子为给定次数多项式的有理函数的逼近。我们分别比较系统地研究了分母为实系数多项式和正系数多项式两种不同情形下的有理逼近的L~p逼近速度估计问题、点态估计问题、共正逼近问题等有重要意义的问题。无论是所得结果还是所用的方法,都是对已有结果和方法的改进和创新。进一步,我们还提出了一些有待解决的问题,这些问题的解决需要用新的方法,但任何正面或反面的回答都将推动这个方向的进一步研究。
第二部分:Turán不等式。我们首先考虑了分母为具有给定奇点的有理函数的Turán不等式,所得结论是Min[Min]的本质性改进,并回答了Min所提出的问题。我们还考虑了加双倍权、A~*权等具有内部奇性权的Turán不等式,对已有方法进行了简化。
第三部分:Fourier分析中若干经典结论的推广。我们在Le和Zhou所提出的GBV条件的基础上,引入了一种新的NBV条件,从而实现了把单调性条件从“单边控制”向“双边控制”的转变。我们研究了MS,QMDS,RBVS,AMDS,GBVS和NBVS等不同数列之间的关系,在此基础上,我们给出了NBV条件在研究三角级数的一致收敛性、L~1收敛性、L~p可积性、连续函数的强逼近等Fourier分析经典问题上的应用,对一些经典的结论作了推广。
The present thesis consists of three parts:Part Ⅰ. Approximation by rational functions with polynomials of real coefficients or positive coefficients as the denomintors and with prescribed numerator degree. We study some important topics such as the L_p-approximation, pointwise estimate and copositive approximation. The results obtained and the methods are interesting and have their values in theories and applications. We also raise some problems for further study. We believe that it needs some new ideas and methods to further development.Part Ⅱ. Turan type inequalities. First, we establish Turan type inequalities for rational functions with prescribed poles, and essentially improve the results of [Min], thus give a positive answer to a problem of Min. Secondly, we establish Turan type inequalities with doubling weights and A* weights. As we know, these are weights having inner singularities. Our method is also an simplification of the classic cases.Part Ⅲ. Improvement of some classical results in Fourier analysis. First, based on the condition of GBV raised by Zhou and Le, we suggest a new kind of condition-XBV condition, which transforms the monotonicity from "one-sided" to "two-sided". We study the relations among MS, QMDS, RBVS, AMDS, GBVS and NBVS. Finally. we improve some classic results in Fourier analysis such as uniform convergence, L~1 convergence, L~p-integrability, strong approximation of continuous functions et al., by applying this new condition.
引文
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