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Como una continuación de la asignatura de análisis funcional, en el siguiente trabajo se realizará un estudio exhaustivo de la teoría espectral de operadores. Se estudiarán y definirán los conceptos básicos de espectro, autovalor y autovector, así como las propiedades que presentan. Además, se introducirán dos tipos de operadores, los operadores autoadjuntos y los operadores compactos. Este tipo de operadores aportarán una gran cantidad de propiedades espectrales y la unión de ambos, nos permitirá establecer una generalización de la diagonalización de matrices. En resumidas cuentas, veremos que todo operador autoadjunto y compacto definido sobre un espacio de Hilbert, permite obtener una base del espacio constituida por sus autovectores y además, dicho operador puede ser “diagonalizado”. Finalmente, este trabajo no es únicamente una generalización de resultados clásicos de álgebra lineal. También es aplicable a la formalización de métodos para la resolución de ecuaciones en derivadas parciales (EPDs). Un ejemplo claro es el método de separación de variables introducido por Fourier.

For any dilation matrix with integral entries A ∈ Rd×d, d ≥ 1, we construct two families of Parseval wavelet frames in L2(Rd). Both families have compact support and any desired number of vanishing moments. The first family has | detA| generators. The second family has any desired degree of regularity. For the members of this family, the number of generators depends on the dilation matrix A and the dimension d, but never exceeds | detA| + d. Our construction involves trigonometric polynomials developed by Heller to obtain refinable functions, the Oblique Extension Principle, and a slight generalization of a theorem of Lai and Stöckler.

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.

Let A be an expansive linear map on Rd preserving the integer lattice and with |detA|=2. We prove that if there exists a self-affine tile set associated to A, there exists a compactly supported wavelet with any desired number of vanishing moments and some symmetry. We put emphasis on construction of wavelets associated to a linear map A on R2 and to the Quincunx dilation on R3 because we can remove the hypothesis of the existence of the self-affine tile set. Our construction is based on low pass filters by Han in dimension one with the dyadic dilation and multiresolution theory. Finally, for some particular dilation matrices, we realize that unidimensional Daubechies low pass filers can be adapted to obtain compactly supported wavelets with any desired degree of regularity and any fix number of vanishing moments.

We give a characterization of all Parseval wavelet frames arising from a given frame multiresolution analysis. As a consequence, we obtain a description of all Parseval wavelet frames associated with a frame multiresolution analysis. These results are based on a version of Oblique Extension Principle with the assumption that the origin is a point of approximate continuity of the Fourier transform of the involved refinable functions. Our results are written for reducing subspaces.

La investigación a realizar durante la duración de la red será, la de buscar los diferentes y principales problemas que, los alumnos encuentran en las principales nociones introducidas durante el curso, así como, los errores más comunes realizados por los alumnos en las diferentes pruebas de evaluación continua desarrollada por la signatura en cuestión. También se realizarán trabajos bibliográficos para proponer nuevos ejercicios/problemas para que los alumnos puedan mejorar la comprensión de los conceptos tratados. Además, se realizarán resúmenes escritos con las diferentes formas de resolución planteadas por los alumnos para que éstos puedan ver las diferentes formas de poder abarcar cada uno de los problemas planteados en la evaluación continua.

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.

Erratum to: J Fourier Anal Appl DOI 10.1007/s00041-015-9442-x

We give a simple orthonormal wavelet system associated to the quincunx matrix. That system has four generators and those generators are sum of two constant functions on triangles. Our construction is based on the structure of a multiresolution analysis. Moreover, we prove that such a system is complete in Lp(R2), 1 < p < ∞. Finally, we give a filter bank algorithm associated to the proposed wavelet system.

We give a complete characterization of the classes of weight functions for which the higher rank Haar wavelet systems are unconditional bases in weighted norm Lebesgue spaces. Particulary it follows that higher rank Haar wavelets are unconditional bases in the weighted norm spaces with weights which have strong zeros at some points. This shows that the class of weight functions for which higher rank Haar wavelets are unconditional bases is much richer than it was supposed.

Let d ≥1. For any A ∈ Z d×d such that | det A | = 2 , we construct two families of Parseval wavelet frames with two generators. These generators have compact support, any desired number of vanishing moments, and any given degree of regularity. The first family is real valued while the second family is complex valued. To construct these families we use Daubechies low pass filters to obtain refinable functions, and adapt methods employed by Chui and He and Petukhov for dyadic dilations to this more general case. We also construct several families of Parseval wavelet frames with three generators having various symmetry properties. Our constructions are based on the same refinable functions and on techniques developed by Han and Mo and by Dong and Shen for the univariate case with dyadic dila- tions.

We are interested in the problem of when the density condition in a multiresolution analysis defined in Lp(Rn), 1≤p≤∞, holds. Indeed, if 2≤p<∞, we obtain sufficient conditions on the generators of a multiresolution analysis in order to the density condition is satisfied. We emphasis on the requirement of the Fourier transform in a neighborhood of the origin. This involves the notion of density point. When 1≤p≤2, the obtained condition is necessary. Moreover, we study the same problem when a multiresolution analysis is defined in the subspace of L∞(Rn) of the set of all continuous functions vanishing at infinite.

El objetivo principal de esta red es procurar la mejora en la coordinación y el seguimiento de los cursos correspondientes al Grado en Matemáticas de la Facultad de Ciencias de la Universidad de Alicante, que se engloba dentro del proceso general del seguimiento de todos los títulos de la Facultad de Ciencias. Para lograr este objetivo se han planteado varias tareas entre las que destacamos: el estudio de las recomendaciones recibidas en el informe final de reacreditación del título emitido por la AVAP (Agencia Valenciana de Evaluación y Prospectiva) tras la renovación de la acreditación del Grado durante el curso anterior con una valoración global FAVORABLE, estudio de la evolución de los resultados de las convocatorias ordinarias, encuestas a todos los alumnos sobre el funcionamiento de cada asignatura, encuesta a alumnos de 3º y 4º curso sobre Másters y, por último, estudio comparativo de tasas de abandono con los grados de matemáticas del resto de universidades españolas. La red está dirigida por la coordinadora del Grado en Matemáticas y formada por los coordinadores de cada uno de los semestres.

Tal y como se ha desarrollado en años anteriores, el objetivo principal de esta red ha sido procurar la mejora en la coordinación y el seguimiento de los cursos correspondientes al Grado en Matemáticas de la Facultad de Ciencias de la Universidad de Alicante, que en el presente curso académico ha recibido la renovación de la acreditación y se engloba dentro del proceso general del seguimiento de todos los títulos de la Facultad de Ciencias. La red está coordinada por la coordinadora del Grado en Matemáticas y formada por los coordinadores de cada uno de los semestres. Se pretende evidenciar los progresos del título en el desarrollo del Sistema de Garantía Interno de Calidad (SGIC), con el fin de detectar las posibles deficiencias en el proceso de implantación del grado y contribuir a la propuesta de acciones para mejorar su diseño y desarrollo.

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.

En este trabajo mostramos la forma planteada por los miembros de la red para llevar a cabo la evaluación continua de una asignatura de matemáticas impartida en los grados de Química y Geología de la Facultad de Ciencias de la Universidad de Alicante. La idea principal es cambiar las tradicionales clases prácticas de pizarra por parte del profesor por otras estrategias más participativas por parte del alumno. Así además de las clásicas hojas de problemas que el profesor prepara para la resolución por parte del alumno, éstas se combinan con unas prácticas se preparan por parte de los componentes de la red y que son realizadas en clase por parte de los alumnos trabajando en grupos reducidos. Tras su elaboración los profesores puntúan y devuelven dichas prácticas a los alumnos para que puedan notar y examinar sus errores. La idea es acercar e interactuar de manera constante entre el alumno y el profesor así como realizar una evaluación continua basada en una gran cantidad de información.

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.

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.

In this paper we study the dependence of the first eigenvalue of the infinity Laplace with respect to the domain. We prove that this first eigenvalue is continuous under some weak convergence conditions which are fulfilled when a sequence of domains converges in Hausdorff distance. Moreover, it is Lipschitz continuous but not differentiable when we consider deformations obtained via a vector field. Our results are illustrated with simple examples.

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.

For any 2 x 2 dilation matrix with integer entries and |det A| = 2, we construct a family of smooth compactly supported tight wavelet frames with three generators in L2(R2). Our construction involves some compactly supported refinable functions, the oblique extension principle, and a slight generalization of a theorem of Lai and Stöckler. Estimates for the degrees of smoothness are given. With the exception of a polynomial whose coefficients must in general be computed by spectral factorization, the framelets are expressed in closed form in the frequency domain, in terms of elementary transcendental functions. By means of two examples we also show that for low degrees of smoothness the use of spectral factorization may be avoided.