Approximation Theory and Complex Dynamics

逼近理论和复杂动力学

• 批准号: 2246876
• 负责人: Kirill Lazebnik
• 金额: $ 18.03万
• 依托单位: University of North Texas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246876&HistoricalAwards=false
• 关键词: Approximation; Theory; Complex; Dynamics

This project involves the study of approximation theory in the setting of complex functions, with applications to complex dynamics. Approximation theory seeks to understand the extent to which the behavior of a general function can be effectively modeled by that of functions drawn from a more restricted class. Efficient approximation of functions is of relevance for numerical calculation. Since the only calculations that can be carried out numerically are the elementary operations of addition, subtraction, multiplication, and division, in practical terms it is of importance to understand when the values of general functions are well approximated by the values of either polynomial or rational functions. In many situations, the values of the approximant resemble those of the general function only for a sampling of input values. What can be said about values of the approximant for other choices of input? This is the main question studied in this project, with the following application in mind: when a general function is iterated to produce a dynamical system, to what extent does the dynamical behavior of an approximant resemble the dynamical behavior of the original function? The project will also contribute to the development of human resources through educational outreach at the high school level as well as mentoring and training at the undergraduate and graduate levels, and will facilitate the interaction of different fields of mathematics through the organization of conferences and seminars.The Principal Investigator will study the approximation of analytic functions in one complex variable by polynomials, rational functions, and transcendental entire functions. Quasiconformal mappings will be a major tool in this study. The quasiconformal approach to this particular subject is largely unexplored, and affords control over geometric properties of the approximants such as location of critical points and critical values. Such control can be used to understand the intricate dynamics recently proven to exist for transcendental entire functions. Another anticipated application lies in an improved understanding of the geometries of lemniscates (level sets of polynomials or rational functions) which relate to the emerging field of pattern recognition as a tool to distinguish between different planar shapes. The PI also will investigate whether a better understanding of the geometry of polynomial or rational approximants yields new insight on the numerical implementation of root-finding algorithms which rely essentially on such approximants. Interactions between the different fields alluded to above (approximation theory, geometric function theory, complex dynamics, and numerical analysis) will be fostered via the organization of conferences, meetings, and seminars.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

本课题研究复杂函数的近似理论，并将其应用于复杂动力学。近似理论试图理解一般函数的行为在多大程度上可以通过从更有限的类中提取的函数的行为来有效地建模。函数的有效逼近与数值计算有关。由于只有加法、减法、乘法和除法这些基本运算才能在数值上进行计算，所以在实际操作中，理解什么时候一般函数的值与多项式函数或有理函数的值很接近是很重要的。在许多情况下，近似的值与一般函数的值相似，只是对于输入值的采样。对于其他输入选项的近似值可以说些什么呢？这是本项目研究的主要问题，考虑到以下应用：当一个一般函数迭代产生一个动力系统时，近似值的动力行为在多大程度上与原始函数的动力行为相似？该项目还将通过在高中一级的教育推广以及在本科和研究生一级的指导和培训来促进人力资源的发展，并将通过组织会议和研讨会促进不同数学领域的相互作用。首席研究员将研究用多项式、有理函数和超越整函数逼近解析函数在一个复变量中的近似。拟共形映射将是本研究的主要工具。这个特定主题的拟共形方法在很大程度上尚未被探索，并且提供了对近似的几何性质的控制，例如临界点和临界值的位置。这种控制可以用来理解最近被证明存在于超越整体函数的复杂动力学。另一个预期的应用在于提高对lemmniscates（多项式或有理函数的水平集）几何的理解，这与模式识别的新兴领域有关，作为区分不同平面形状的工具。PI还将研究更好地理解多项式或有理近似的几何是否会对基本上依赖于这些近似的求根算法的数值实现产生新的见解。以上提到的不同领域（近似理论、几何函数理论、复杂动力学和数值分析）之间的相互作用将通过组织会议和研讨会来促进。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。