# Applied Functional Analysis (15w5088)

Arriving in Oaxaca, Mexico Sunday, June 28 and departing Friday July 3, 2015

## Organizers

Feng Dai (University of Alberta )

(Texas A & M University)

Sergey Konyagin (Steklov Mathematical Institute of Russian Academy of Sciences)

Vladimir Temlyakov (University of South Carolina)

(ICREA + Centre de Recerca Matemàtica, Barcelona)

## Objectives

The objective of the workshop is to provide a unique forum for exchanging ideas and in-depth discussions on different aspects of recent developments in applied functional and harmonic analysis and related areas, and to encourage outstanding mathematical contributions in these areas.

Leading experts and promising young researchers will be brought together to tackle some challenging open problems, and to discuss future research directions in applied harmonic analysis and constructive approximation.

Format:
Each day there will be several plenary lectures given by top experts, and one shorter session on open problems which will be prepared by two participants. There will also be time for informal discussions.

The proposed workshop will cover several topics in applied harmonic analysis, constructive approximation theory, numerical and functional analysis, which represent research interests of the majority of the participants. These topics can be divided into the following two closely related parts:

1. Nonlinear approximation.

The goal is to carry out fundamental mathematical and algorithmic study to significantly increase our ability to process (compress, denoise, etc.) large data sets. Many multi-parameter real-world problems are modeled on spaces of multivariate functions in $n$ variables, where $n$ may be very large. Since these problems can almost never be solved analytically, one is interested in suitable model assumptions and good approximate solutions within a reasonable computing time.

The main technique that will be used in achieving this goal is based on nonlinear sparse representations. Sparse representations of a function are not only a powerful analytic tool but they are utilized in many application areas such as image/signal processing and numerical computation. The backbone of finding sparse representations is the concept of $m$-term approximation of the target function by the elements of a given system of functions (dictionary). Since the elements of the dictionary used in the $m$-term approximation are allowed to depend on the function being approximated, this type of approximation is very efficient.

The fundamental question is how to construct good methods (algorithms) of approximation. The purpose of such research is to design and study general nonlinear methods of approximation that are practically implementable. In particular, specific methods of approximation that belong to a family of greedy approximation methods are of special interest. These methods allow us to build sparse representations in an economical way.

The theory of greedy approximations have been rapidly developing recently, thus the focus will be made on quantitative aspects of greedy approximations.

2. Approximation and Fourier analysis on multivariate domains.

Approximation and Fourier analysis on multivariate domains is a big and rapidly developing research area. It has turned out to be a very useful tool in many applications, such as numerical analysis, learning theory, compressed sensing, signal processing, biomedical optics, and geographic information systems. Much of the recent research has been focused on analysis on the unit sphere and other related domains such as the unit ball and the simplex. The importance of the research on these domains lies in the fact that the sphere is a natural setting for many mathematical models and therefore analysis on sphere-related domains are often encountered in many practical applications and
many different mathematical branches. However, due to the difficult geometry and topology of the underlying domains, many important approximation problems in this area remain open. The understanding of these problems requires a set of new ideas and tools and often reveals the underlying structure of the object under study, yielding valuable insights for further applications.