On Schauder Bases Properties of Multiply Generated Gabor Systems

Let $A$ be a finite subset of $L^2(\mathbb{R})$ and $p,q\in\mathbb{N}$. We characterize the Schauder basis properties in $L^2(\mathbb{R})$ of the Gabor system $$G(1,p/q,A)=\{e^{2\pi i m x}g(x-np/q) : m,n\in \mathbb{Z}, g\in A\},$$ with a specific ordering on $\mathbb{Z}\times \mathbb{Z}\times A$. The characterization is given in terms of a Muckenhoupt matrix $A_2$ condition on an associated Zibulski-Zeevi type matrix.Comment: 14 page

Frame Theory for Signal Processing in Psychoacoustics

This review chapter aims to strengthen the link between frame theory and signal processing tasks in psychoacoustics. On the one side, the basic concepts of frame theory are presented and some proofs are provided to explain those concepts in some detail. The goal is to reveal to hearing scientists how this mathematical theory could be relevant for their research. In particular, we focus on frame theory in a filter bank approach, which is probably the most relevant view-point for audio signal processing. On the other side, basic psychoacoustic concepts are presented to stimulate mathematicians to apply their knowledge in this field

Operator-Valued Frames Associated with Measure Spaces

abstract: Since Duffin and Schaeffer's introduction of frames in 1952, the concept of a frame has received much attention in the mathematical community and has inspired several generalizations. The focus of this thesis is on the concept of an operator-valued frame (OVF) and a more general concept called herein an operator-valued frame associated with a measure space (MS-OVF), which is sometimes called a continuous g-frame. The first of two main topics explored in this thesis is the relationship between MS-OVFs and objects prominent in quantum information theory called positive operator-valued measures (POVMs). It has been observed that every MS-OVF gives rise to a POVM with invertible total variation in a natural way. The first main result of this thesis is a characterization of which POVMs arise in this way, a result obtained by extending certain existing Radon-Nikodym theorems for POVMs. The second main topic investigated in this thesis is the role of the theory of unitary representations of a Lie group G in the construction of OVFs for the L^2-space of a relatively compact subset of G. For G=R, Duffin and Schaeffer have given general conditions that ensure a sequence of (one-dimensional) representations of G, restricted to (-1/2,1/2), forms a frame for L^{2}(-1/2,1/2), and similar conditions exist for G=R^n. The second main result of this thesis expresses conditions related to Duffin and Schaeffer's for two more particular Lie groups: the Euclidean motion group on R^2 and the (2n+1)-dimensional Heisenberg group. This proceeds in two steps. First, for a Lie group admitting a uniform lattice and an appropriate relatively compact subset E of G, the Selberg Trace Formula is used to obtain a Parseval OVF for L^{2}(E) that is expressed in terms of irreducible representations of G. Second, for the two particular Lie groups an appropriate set E is found, and it is shown that for each of these groups, with suitably parametrized unitary duals, the Parseval OVF remains an OVF when perturbations are made to the parameters of the included representations.Dissertation/ThesisDoctoral Dissertation Mathematics 201

Applied Harmonic Analysis and Data Science (hybrid meeting)

Data science has become a field of major importance for science and technology nowadays and poses a large variety of challenging mathematical questions. The area of applied harmonic analysis has a significant impact on such problems by providing methodologies both for theoretical questions and for a wide range of applications in signal and image processing and machine learning. Building on the success of three previous workshops on applied harmonic analysis in 2012, 2015 and 2018, this workshop focused on several exciting novel directions such as mathematical theory of deep learning, but also reported progress on long-standing open problems in the field

Basis expansions in applied mathematics

Basis expansions are an extremely useful tool in applied mathematics. By using them, we can express a function representing a physical quantity as a linear combination of simpler ``modules'' with well-known properties. They are particularly useful for the applications described in this thesis. Perhaps the best known expansion of this type is the Fourier series of a periodic function, as decomposition into the infinite sum of simple sinusoidal and cosinusoidal elements, originally proposed by Fourier to study heat transfer. This dissertation employs some mathematical tools on problems taken from various areas of Engineering, exploiting their expansion properties: 1) Non-integer bases, which are applied to mathematical models in Robotics (Chapter 2). In this Chapter we study, in particular, a model for snake-like robots based on the Fibonacci sequence. It includes an investigation of the reachableworkspace, a more general analysis of the control system underlying the model, its reachability and local controllability properties. 2) Orthonormal bases, Riesz bases: exponential and cardinal Riesz basis with perturbations (Chapter 3). In this Chapter we obtain also a stability result for cardinal Riesz basis in the case of complex perturbations of the integers. We also consider a mathematical model for energy of the signal at the output of an ideal DAC, in presence of sampling clock jitter. When sampling clock jitter occurs, the energy of the signal at the output of ideal DAC does not satisfies a Parseval identity. Nevertheless, an estimation of the signal energy is here shown by a direct method involving cardinal series. 3) Orthogonal polynomials (Chapter 4). In this Chapter we introduce a new sequence of polynomials, which follow the same recursive rule of the well-known Lucas-Lehmer integer sequence. We show the most important properties of this sequence, relating them to the Chebyshev polynomials of the first and second kind. We discuss some relations between zeros of Lucas-Lehmer polynomials and Gray code. We study nested square roots of 2 applying a "binary code" that associates bits 0 and 1 to + and - signs in the nested form. This gives the possibility to obtain an ordering for the zeros of Lucas-Lehmer polynomials, which take the form of nested square roots of 2. These zeros are used to obtain two new formulas for Pi

Studies on electromagnetic turbulence and edge phenomena in fusion plasmas

The magnetic well depth if one of the principal actors when the stability of a confined plasma is analysed. It is the main stabilising mechanism in the TJ-II stellarator, as this is an almost shearless device. This and TJ-II's ability for changing the currents of its coils make the Spanish stellarator a perfect candidate for magnetic well studies. This thesis presents an exhaustive study on plasma performance and stability under theoretically unstable magnetic well conditions. NBI-heated reproducible plasmas were successfully produced even for the most stability adverse conditions and a link between the Alfén Eigenmodes and magnetic well depth was found. Visible light emission at the plasma edge of the JET tokamak has been studied with an intensified fast visible camera since the installation of its ITER-Like Wall. A method to characterize the evolution of ELMs in the divertor and relate the recorded signal with other diagnostics at JET has been developed. A large Matlab library orientated to treat and share the data produced by the intensified fast visible camera has been made available to the users of this diagnostic.Programa Oficial de Doctorado en Plasmas y Fusión NuclearPresidente: Enrique Ascasibar Zubizarreta.- Secretario: Luis García Gonzalo.- Vocal: Antonio López Fragua