Density order of Parseval wavelet frames from extension principles

We characterize approximation order and density order of those Parseval wavelet frames obtained from Oblique Extension Principle. These notions are closely related to approximation order and density order by a quasi-projection operator. To give our characterizations, we shall explain the behavior on a neighborhood of the origin of the Fourier transform of a refinable function. In particular, we invoke the classical notion of approximate continuity. We write our results in the multivariate context of Parseval wavelet frames associated to A, an expansive linear map preserving the integer lattice

The Lebesgue differentiation theorem revisited

We prove a general version of the Lebesgue differentiation theorem where the averages are taken on a family of sets that may not shrink nicely to any point. These families of sets involve the unit ball and its dilated by negative integers of an expansive linear map. We also give a characterization of the Lebesgue measurable functions on R^n in terms of approximate continuity associated to an expansive linear map

Caracterización y propiedades de las funciones de escala y filtros de paso bajo de un análisis multirresolución

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura 21-09-200

Some smooth compactly supported tight framelets associated to the quincunx matrix

We construct several families of tight wavelet frames in L2(R2)L2(R2) associated to the quincunx matrix. A couple of those families has five generators. Moreover, we construct a family of tight wavelet frames with three generators. Finally, we show families with only two generators. The generators have compact support, any given degree of regularity, and any fixed number of vanishing moments. Our construction is made in Fourier space and involves some refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. In addition, we will use well known results on construction of tight wavelet frames with two generators on RR with the dyadic dilation. The refinable functions we use are constructed from the Daubechies low pass filters and are compactly supported. The main difference between these families is that while the refinable functions associated to the five generators have many symmetries, the refinable functions used in the construction of the others families are merely even.The first author was partially supported by MEC/MICINN grant #MTM2011-27998 (Spain)

A family of nonseparable scaling functions and compactly supported tight framelets

Given integers b and d, with d>1 and |b|>1, we construct even nonseparable compactly supported refinable functions with dilation factor bb that generate multiresolution analyses on L2(Rd). These refinable functions are nonseparable, in the sense that they cannot be expressed as the product of two functions defined on lower dimensions. We use these scaling functions and a slight generalization of a theorem of Lai and Stöckler to construct smooth compactly supported tight framelets. Both the refinable functions and the framelets they generate can be made as smooth as desired. Estimates for the supports of these refinable functions and framelets, are given.The first author was partially supported by Spanish Science Ministry grant JC2010-0012

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Let A∈Rd×d, d≥1 be a dilation matrix with integer entries and |detA|=2. We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is 2d+d and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. For the particular case d=2 and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values.The first author was partially supported by MEC/MICINN Grant #MTM2011-27998 (Spain)

Upper bounds for the decay rate in a nonlocal p-Laplacian evolution problem

We obtain upper bounds for the decay rate for solutions to the nonlocal problem ∂tu(x,t)=∫RnJ(x,y)|u(y,t)−u(x,t)|p−2(u(y,t)−u(x,t))dy with an initial condition u0∈L1(Rn)∩L∞(Rn) and a fixed p>2. We assume that the kernel J is symmetric, bounded (and therefore there is no regularizing effect) but with polynomial tails, that is, we assume a lower bounds of the form J(x,y)≥c1|x−y|−(n+2σ), for |x−y|>c2 and J(x,y)≥c1, for |x−y|≤c2. We prove that ∥u(⋅,t)∥Lq(Rn)≤Ct−n(p−2)n+2σ(1−1q) for q≥1 and t large.This work was partially supported by MEC MTM2010-18128 and MTM2011-27998 (Spain)

Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length

We study local (the heat equation) and nonlocal (convolution-type problems with an integrable kernel) evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions to both local and nonlocal problems is given by the solution of the heat equation, but on a starshaped graph in which there are only one node and as many infinite edges as in the original graph. In this way, we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single point when looking at the asymptotic behaviour of the solutions. For this star-shaped limit problem, the asymptotic behaviour of the solutions is just given by the solution to the heat equation in a half line with a Neumann boundary condition at x = 0 and initial datum (2M/N)δx=0 where M is the total mass of the initial condition for our original problem and N is the number of edges of infinite length. In addition, we show that solutions to the nonlocal problem converge, when we rescale the kernel, to solutions to the heat equation (the local problem), that is, we find a relaxation limit.L. I. was partially supported by a grant of Ministry of Research and Innovation, CNCSUEFISCDI, project PN-III-P1-1.1-TE-2016- 2233, within PNCDI III. J.D.R. partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), PICT-2018-03183 (Argentina) and UBACyT grant 20020160100155BA (Argentina)

El laboratorio de matemáticas como estrategia docente

En esta experiencia docente se pone en práctica una forma diferente de llevar a cabo las clases prácticas de algunas asignaturas de matemáticas de primer curso de la Facultad de Ciencias de la Universidad de Alicante. El objetivo es sustituir las habituales clases prácticas, donde el profesor realiza los ejercicios en la pizarra, por la resolución de problemas por parte de los alumnos incorporando además otras estrategias docentes; es decir, además de las hojas de ejercicios que el profesor prepara para los alumnos, los docentes preparan unas actividades prácticas para que sean realizadas en clase por los estudiantes, en grupos reducidos y guiados por el profesor. Estas actividades son puntuadas por el tutor y, tras ser devueltas a los alumnos, éstos deberán observar y analizar sus errores con la ayuda extra de las tutorías presenciales y virtuales. Con este método se consigue una mayor interacción entre alumno y profesor, un estudio continuo de la asignatura y una constante evaluación del profesor al alumno y del alumno a la asignatura

Decay estimates for nonlinear nonlocal diffusion problems in the whole space

In this paper, we obtain bounds for the decay rate in the L r (ℝ d )-norm for the solutions of a nonlocal and nonlinear evolution equation, namely, ut(x,t)=∫RdK(x,y)|u(y,t)−u(x,t)|p−2(u(y,t)−u(x,t))dy,x∈Rd,t>0. . We consider a kernel of the form K(x, y) = ψ(y−a(x)) + ψ(x−a(y)), where ψ is a bounded, nonnegative function supported in the unit ball and a is a linear function a(x) = Ax. To obtain the decay rates, we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T(u)=−∫RdK(x,y)|u(y)−u(x)|p−2(u(y)−u(x))dy,1⩽p<∞. . The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole space ℝ d : λ1,p(Rd)=2(∫Rdψ(z)dz)∣∣∣∣1|detA|1/p−1∣∣∣∣p. Moreover, we deal with the p = ∞ eigenvalue problem, studying the limit of λ 1,p 1/p as p→∞.L. I. Ignat is partially supported by grants PN II-RU-TE 4/2010 and PCCE-55/2008 of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, MTM2011-29306-C02-00, MICINN, Spain and ERC Advanced Grant FP7-246775 NUMERIWAVES. D. Pinasco is partially supported by grants ANPCyT PICT 2011-0738 and CONICET PIP 0624. J. D. Rossi and A. San Antolin are partially supported by the grant MTM2011-27998 MICINN MICINN, Spain