Harmonic Analysis, Sparse Approximations and Applications

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Akram Aldroubi (Vanderbilt University)          (view abstract in pdf)
Time-Frequency Uncertainty Principles For Shift Invariant Spaces

The Balian-Law Theorem states that if a Gabor system generated by a function g is Riesz basis for L²(R), then g cannot be well localized in both time and frequency. Specifically, ||xg(x)||₂ ||ω^g(ω)||₂ = ∞. For shift invariant spaces, time-frequency obstructions also occur. For example, we will show that if φ generates a principal shift invariant space which is also 1/n invariant for some n>1, then φ cannot be well localized in both time and frequency. For example, ||xg(x)||₂ = ∞. We will present these time frequency obstructions for φ and show that the results are optimal.

 

Esteban Andruchow  (Universidad Nacional de General Sarmiento - IAM - CONICET)      (view abstract in pdf)
The rectifiable distance in the unitary Fredholm group

Let U_c(H)={u: u unitary and u-1 compact} stand for the unitary Fredholm group. We prove the following convexity result. Denote by d_∞ the rectifiable distance induced by the Finsler metric given by the operator norm in U_c(H). If u₀,u₁,u are in U_c(H) and the geodesic β joining u₀ and u₁ in U_c(H) verify d_∞(u,β)<π/2, then the map f(s)=d_∞(u,β(s)) is convex for s in [0,1]. In particular the convexity radius of the geodesic balls in U_c(H) is π/4. The same convexity property holds in the p-Schatten unitary groups U_p(H)={u: u unitary and u-1 in the p-Schatten class}, for p an even integer, p≥4 (in this case, the distance is strictly convex). The same results hold in the unitary group of a C^*-algebra with a faithful finite trace. We apply this convexity result to establish the existence of curves of minimal length with given initial conditions, in the unitary orbit of an operator, under the action of the Fredholm group. We characterize self-adjoint operators A such that this orbit is a submanifold (of the affine space A+K(H), where K(H)=compact operators).

 

John Benedetto (University of Maryland)     (view abstract in pdf)
Frames for wavelet sets and classification

The theme is the role of frames in providing effective tools to deal with large data sets. There are two case studies. The mathematical tools are wavelet theory, Fourier analysis, and frame potential energy analysis.

The first case constructs simple, smooth dyadic wavelet frames for Euclidean space from ONE wavelet. A surprising phenomenon, called a frame bound gap arises; and these gaps are analyzed and computed.

The second case designs a classification algorithm, where frames are required to balance classification with dimension reduction. The technology naturally combines frame potential energy with discrete Wiener amalgam spaces. Examples include the analysis of hyperspectral and retinal data.

 

Ricardo Durán (Universidad de Buenos Aires - CONICET)     (view abstract in pdf)
Solutions of the divergence and analysis of the Stokes equations

In this talk we present some results on the analysis of the Stokes equations which model the displacement of an incompressible viscous fluid.

First, we recall the variational analysis which allows to prove the well posedness of the Stokes equations in bounded Lipschitz domains. The main tool to obtain this result is
the so called inf-sup condition, which is related to the existence of solutions of the divergence in appropriate Sobolev spaces.

In the second part of the talk we show some simple examples of non-Lipschitz domains (namely, cuspidal domains) where the standard inf-sup condition is not valid. Moreover, we show that, in these domains, the Stokes equations are not well posed in the standard Sobolev spaces. Finally, we show how the variational analysis can be generalized to prove the well posedness of the Stokes equations in cuspidal domains in appropriate weighted Sobolev spaces.

 

Liliana Forzani (Universidad Nacional del Litora - IMAL - CONICET)     (view abstract in pdf)
You can still use PCA (sometimes)

Starting with its definition in PCA, I will explain the advantages of using sufficient dimension reduction instead of principal component analysis when we want to discriminate populations. But later in the talk I will give some conditions under which PCA (the most commonly used technique of reduction) is appropriate for discrimination.

This is a joint work with R. Dennis Cook

 

Pola Harboure (Universidad Nacional del Litoral - Instituto de Matemática Aplicada del Litoral)     (view abstract in pdf)
Harmonic Analysis related to Schrödinger operators

Let us consider the Schrödinger operator on R^d, d≥3, L=−∆+V where the potential V≥0 is a function satisfying, for some q>d/2 , the reverse Hölder inequality

( 1/|B|  ∫_BV(y)^q dy )^1/q  ≤ C/|B|  ∫_BV(y) dy

for every ball B in R^d.

The general theory of semigroups, in particular Yosida’s generating Theorem, implies that L is the infinitesimal operator of a semigroup, formally denoted by T_t=e^−tL, that solves the diffusion problem

^d/_dt u(·, t) = −Lu(·, t),
u(·, 0) = f,

by setting u(x, t)=e^−tLf(x).
In this talk we will introduce the main operators of the Harmonic Analysis in this context and we will make a review of their behavior on the L^p spaces, pointing out the similarities and the differences with the classical versions corresponding to the Laplacian.

 

Joos Heintz  (Universidad de Buenos Aires - CONICET)     (view abstract in pdf)
Easy polynomials which are hard to interpolate
(Joint work with N. Giménez, G. Matera and P. Solernó)

In this talk we introduce and discuss a new computational model for Hermite--Lagrange interpolation with non-linear classes of polynomial interpolants. We distinguish between an interpolation problem and an algorithm that solves it. Our model includes also coalescence phenomena and captures a large variety of known Lagrange--Hermite interpolation problems and algorithms. Like in traditional Hermite-Lagrange interpolation, our model is based on the execution of arithmetic operations (including divisions) in the field where the data (nodes and values) are interpreted and arithmetic operations are counted at unit costs. This leads us to a new view of rational functions and maps defined on arbitrary constructible subsets of complex affine spaces. For this purpose we have to develop new tools in algebraic geometry which themselves are mainly based on Zariski's Main Theorem and the theory of places (or equivalently: valuations). We finish this talk by exhibiting two examples of Lagrange interpolation problems with non-linear classes of interpolants, which do not admit efficient interpolation algorithms (one of these interpolation problems requires even an exponential quantity of arithmetic operations in terms of the number of the given nodes in order to represent some of the interpolants). In other words, classic Lagrange interpolation algorithms are asymptotically optimal for the solution of these selected interpolation problems and nothing is gained by allowing interpolation algorithms and interpolation classes to be non--linear. We show also that classic Lagrange interpolation algorithms are almost optimal for generic nodes and values. This generic data cannot be substantially compressed by using non-linear techniques.

 

Eugenio Hernández (Universidad Autónoma de Madrid)      (view abstract in pdf)
The role of democracy functions in Approximation Theory  (abstract)

We prove optimal embeddings for nonlinear approximation spaces A_q^α, in terms of weighted Lorentz sequence spaces, with the weights depending on the democracy functions of the basis. As applications we recover known embeddings for N-term
wavelet approximation in L^p, Orlicz, and Lorentz norms.

 

Ricardo Miró (Corte Suprema de Justicia de la Nación - Cuerpo de Peritos Oficiales)     (view abstract in pdf)
Evaluación de la Calidad Institucional de una República con Elementos
de la Teoría de Juegos

El dilema del prisionero, repetido indefinidamente, permite sentar las bases para medir la calidad institucional de un Estado Republicano. En un sentido no excluyente, éste puede subordinar su desarrollo a una Constitución, o bien a los intereses de sectores corporativos. Bajo ciertas condiciones que emergen de la estrategia de Axelrod-Rapoport, es posible vislumbrar un modelo de evaluación en el tiempo para el proceso global.

 

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