Mathematical Sciences: Multivariate Approximation

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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.