Mathematical Sciences: Exponential Polynomials, Lacunary Series; Weighted Norm Inequalities

数学科学：指数多项式、空位级数；

• 批准号: 9706775
• 负责人: Fedor Nazarov
• 金额: $ 3.88万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706775&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Exponential; Polynomials; Lacunary

ABSTRACT Nazarov Nazarov will investigate exponential polynomials and lacunary series and on weighted norm inequalities. The objectives of the proposed research are: 1) to get a complete version of the Turan Lemma, 2) to investigate the distribution of values of trigonometric polynomials with random spectra, 3) to complete the theory of weighted norm inequalities for singular integral operators with one matrix weight and 4)to start an attack on the scalar two-weight case. The problems described in this proposal are of fundamental character and have numerous connections to other fields some of which in their turn have a direct impact on science and technology. For example, fast computing and data compression based on wavelets. The idea of the wavelet compression is strikingly simple: you just take the wavelet expansion of your image and drop all small terms, memorizing only several "large" ones. The problem is that you should be sure that the large terms are really few and that the small terms together cannot add up to something huge. The formal requirement (which is more or less just the above phrase translated into mathematical language) is that the system of wavelets should form a basis in a certain function space.

摘要 纳扎罗夫 纳扎罗夫将研究指数多项式和空隙级数以及加权范数不等式。本研究的目标是：1）获得图兰引理的完整版本，2）研究具有随机谱的三角多项式值的分布，3）完成奇异积分算子的加权范数不等式理论一个矩阵权重和 4) 开始对标量两个权重情况进行攻击。 该提案中描述的问题具有根本性，并且与其他领域有许多联系，其中一些领域反过来又对科学和技术产生直接影响。例如，基于小波的快速计算和数据压缩。 小波压缩的想法非常简单：您只需对图像进行小波展开并丢弃所有小项，只记住几个“大”项。 问题是，你应该确保大项确实很少，而小项加起来不能构成巨大的东西。形式要求（或多或少就是上述短语翻译成数学语言）是小波系统应在某个函数空间中形成基。