Complex approximation on Riemann surfaces and in infinite dimensions

黎曼曲面和无限维上的复近似

• 批准号: RGPIN-2016-04107
• 负责人: Gauthier, Paul
• 金额: $ 1.31万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708901
• 关键词: Complex; approximation; Riemann; surfaces; infinite

The topic of this research is approximation theory. It is theoretical in nature, which means that it is potentially applicable to a wide variety of situations, but in order to apply to a particular field, say energy, collaboration will be required between applied mathematicians, physcists and engineers in order to make explicit how such mathematical entities as a, b and c can be interpreted as say force, potential energy and kinetic energy. More specifically, my research involves trying to approximate given functions by nicer ones. The functions we encounter in the world are often too complicated to deal with, so we seek simpler mathematical functions which approximate the real functions of the world. If these simpler functions approximate the real functions well enough, we can make the desired predictions within an acceptable degree of accuracy. For example, if we can make a watch which loses only one second in one hundred years, most people would be satisfied. Such a watch only approximates real time, but extremely well. The nice functions I wish to approximate with are the so-called analytic functions. These are functions f(x), which have the virtue that they can be written in the form of infinite series f(x) = a + bx + cx2 + . . . where x is the variable and the coefficients a, b, c, . . . are constants. Analytic functions have extremely nice properties. For example, their global behaviour is uniquely determined by their local behavior. This means, for example, that if we know that some universal phenomenon behaves analytically, then, in principle, its behavior behind some distant star is completely determined by its behaviour on earth. Thus, (again in principle) we can know the behavior behind the star without going there, provided we can know it sufficiently well right here on earth. Our approach to approximating a given function g by an analytic function f is a localization process. We first approximate the function g in various local regions by analytic functions in these local regions. Then, we try to piece together these local analytic functions in order to get a global analytic function f which approximates our initial function g. This is hard to do, because the aforementioned uniqueness property of analytic functions makes it very difficult to patch analytic functions together and still retain analyticity.

本研究的主题是逼近理论。它本质上是理论的，这意味着它可能适用于各种各样的情况，但为了应用到一个特定的领域，比如能量，将需要应用数学家、物理学家和工程师之间的合作，以明确如何将a、b和c这样的数学实体解释为比方说力、势能和动能。更具体地说，我的研究涉及尝试用更好的函数来逼近给定的函数。我们在世界上遇到的函数往往太复杂，难以处理，所以我们寻求更简单的数学函数，它们近似于世界的真实函数。如果这些更简单的函数足够好地逼近实函数，我们就可以在可接受的精度范围内做出所需的预测。例如，如果我们能制造一块一百年只慢一秒的手表，大多数人都会满意。这样的手表只能接近实时，但非常准确。我想要近似的好函数是所谓的解析函数。这些是函数f(X)，它们的优点是它们可以写成无穷级数的形式 F(X)=a+bx+cx2+。。。 其中x是变量，系数a，b，c，.。。是常量。解析函数具有非常好的性质。例如，它们的全球行为由它们的本地行为独一无二地决定。这意味着，例如，如果我们知道某些普遍现象的行为是解析的，那么，原则上，它在某个遥远恒星后面的行为完全由它在地球上的行为决定。因此，(同样在原则上)我们可以知道恒星背后的行为，而不需要去那里，前提是我们在地球上能够足够好地了解它。 我们用解析函数f逼近给定函数g的方法是一个局部化过程。我们首先用不同局部区域上的解析函数来逼近函数g。然后，我们试图将这些局部解析函数拼凑在一起，以得到一个全局解析函数f，它近似于我们的初始函数g。这是很难做到的，因为前面提到的解析函数的唯一性使得将解析函数拼接在一起并且仍然保持解析性是非常困难的。