Complex approximation on Riemann surfaces and in infinite dimensions

本研究的主题是近似理论。它本质上是理论性的，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，应用数学家、物理学家和工程师之间需要合作，以便明确如何将a、b和c等数学实体解释为力、势能和动能。更具体地说，我的研究包括尝试用更好的函数来近似给定的函数。我们在世界上遇到的函数往往太复杂而难以处理，因此我们寻求更简单的数学函数，以近似世界的真实函数。如果这些简单的函数能很好地近似真实的函数，我们就能在可接受的精度范围内做出预期的预测。例如，如果我们能制造一种手表，每一百年只损失一秒钟，大多数人都会满意的。这样的手表只能接近真实时间，但非常准确。我想近似的函数是所谓的解析函数。这些是函数f(x)它们的优点是可以写成无穷级数的形式

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.