Mathematical Sciences: Multivariate Approximation

摘要提案：DMS-962292 PI：Temlyakov 近似理论是一个快速变化的数学领域。近似的核心问题仍然是发展有效的方法，用更简单的函数代替一般函数。有些方法是很久以前发明的（基于傅立叶求和、泰勒多项式、三角或代数多项式的最佳近似等的方法）。然而，最近几个数值应用的驱动下，逼近理论的方向已经走向非线性和多元近似。这包括非线性m项逼近、小波、脊函数逼近、双线性逼近等较新的学科。这些学科在数值积分、积分方程数值解、图像压缩、神经网络设计等方面都有应用。重点将放在非线性近似方法，如最佳m项近似，度量熵和双线性，以及它们与其他数学和应用领域的相互作用。结合非线性逼近在数值分析中的应用，Temlyakov将研究一些非线性逼近算法，例如“贪婪”算法。 近似理论寻求用简单对象代替复杂对象的方法。这一思想在解决真实的世界问题的许多应用中已被证明是卓有成效的。这些应用包括信号处理、图像压缩、污染物流分析、金融问题（例如抵押贷款债务）以及许多其他应用。作为模型问题之一，考虑图像压缩。以电视屏幕上的图像（图片）为例。为什么我们要近似它？ 在许多情况下，我们无法传输（或存储在计算机存储器中）图像的整个信息，这可能是因为传输一位信息的成本很高或信道容量有限。这正是应用近似理论可以取得丰硕成果的地方。显然，当我们用其近似值替换图像时，我们会失去图像的质量：我们保留的信息越多，我们所拥有的原始图像的近似度就越好。因此，我们有一个相互作用之间的减少信息和近似的质量。我们试图找到这个问题的最佳解决方案。拟议的研究的目的是继续调查的多元函数的近似方法，这是出于这些类型的应用程序。

ABSTRACT Proposal: DMS-962292 PI: Temlyakov Approximation theory is a rapidly changing area of mathematics. The core problem of approximation continues to be the development of efficient methods for replacing general functions by simpler functions. Some methods were invented long ago (methods based on Fourier sums, Taylor polynomials, best approximations by trigonometric or algebraic polynomials, etc.). More recently however, driven by several numerical applications, the directions of approximation theory have moved toward nonlinear and multivariate approximation. This includes the comparatively new subject of nonlinear m- term approximation, wavelets, approximation by ridge functions, bilinear approximation, etc. These have found applications in numerical integration, numerical solution of integral equations, image compression, design of neural networks, and so on. The purpose of this proposed research is to continue the investigations of several areas of multivariate approximation. Emphasis will be placed on nonlinear methods of approximation such as best m-term approximation, metric entropy, and bilinear, as well as their interaction with other fields of mathematics and applications. Keeping in mind the applications of nonlinear approximation in numerical analysis, Temlyakov will study some nonlinear algorithms of approximation, for instance, "greedy" algorithms. Approximation theory seeks ways to replace complicated object by simpler objects. This idea has proved to be fruitful in many applications to the real world problems. Among these applications are signal processing, image compression, analysis of contaminant flow, finance problems (for instance collateralized mortgage obligation), and many other. As one of the model problems, consider image compression. Take for example an image (picture) on a TV screen. Why should we approximate it? In many cases we cannot afford to transmit (or store in a computer memory) the whole information of an image, perhaps because of a high cost for transmission of a bit of information or limited channel capacity. This is exactly the point where an application of approximation theory can be fruitful. Clearly, when we replace an image by its approximant we lose the quality of picture: the more information we keep the better approximation to the original image we have. As a result we have an interplay between the reduction of information and the quality of approximation. We try to find the best (optimal) solution to this problem. The purpose of the proposed research is to continue the investigations of methods of approximation of multivariate functions which are motivated by these types of types of applications.