Mean convergence of Fourier-Dunkl series

In the context of the Dunkl transform a complete orthogonal system arises in a very natural way. This paper studies the weighted norm convergence of the Fourier series expansion associated to this system. We establish conditions on the weights, in terms of the $A_p$ classes of Muckenhoupt, which ensure the convergence. Necessary conditions are also proved, which for a wide class of weights coincide with the sufficient conditions

Bilinear biorthogonal expansions and the Dunkl kernel on the real line

We study an extension of the classical Paley-Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier-Neumann type series as special cases, and it also provides a bilinear expansion for the Dunkl kernel (in the rank 1 case) which is a Dunkl analogue of Gegenbauer's expansion of the plane wave and the corresponding sampling expansions. In fact, we show how to derive sampling and Fourier-Neumann type expansions from the results related to the bilinear expansion for the Dunkl kernel.Comment: 16 page

Higher Order Riesz Transforms for Fourier-Bessel Expansions

In this paper we investigate the Riesz transforms of order d 1, R d v, for Fourier-Bessel expansions. Weighted norm inequalities are proved for this operator. © 2012 Springer Science+Business Media, LLC

The multiplier of the interval [-1, 1] for the Dunkl transform on the real line

We study the boundedness of the multiplier of the interval [- 1, 1] for the Dunkl transform of order - 1 / 2 on weighted Lp spaces, with 1 < p < . In particular, we get that it is bounded from Lp (R, | x |2 + 1 d x) into itself if and only if 4 ( + 1) / (2 + 3) < p < 4 ( + 1) / (2 + 1). © 2006 Elsevier Inc. All rights reserved

Littlewood-Paley-Stein gk-functions for Fourier-Bessel expansions

gk-Functions related to the Poisson semigroup of Fourier-Bessel expansions are defined for each k 1. It is proved that these gk-functions are Calderón-Zygmund operators in the sense of the associated space of homogeneous type, hence their mapping properties follow from the general theory. © 2010 Elsevier Inc. All rights reserved

El sistema de contrato como estrategia metodológica de aprendizaje autónomo de los estudiantes de las titulaciones de magisterio y educación social

La comunicación que presentamos se basa en una experiencia de innovación educativa llevada a cabo con alumnos de las Titulaciones de Magisterio y Educación Social de la Escuela Universitaria de Magisterio de Bilbao y trata de mostrar en qué medida el Contrato de Aprendizaje Autónomo, puede ser un modelo a seguir para la enseñanza-aprendizaje de alumnos/as universitarios. Las exigencias actuales de las enseñanzas universitarias en el marco del Espacio Europeo de Educación Superior (EEES) suponen una modificación de las modalidades y métodos de enseñanza y del papel del profesor. La educación superior de calidad será aquélla que, más allá de transmitir conocimientos, introduzca situaciones de aprendizaje significativas tanto dentro como fuera del aula, haciendo al alumno protagonista y responsable de su propio aprendizaje. El Sistema de Contrato de aprendizaje está basado en los principios constructivistas de la enseñanza-aprendizaje y destaca la implicación del alumno en su formación. Es una metodología universitaria que posibilita la adecuación de los objetivos del aprendizaje a las características y ritmos del alumno. Facilita el aprendizaje autónomo de aquellos alumnos que por distintas razones no pueden asistir regularmente a las clases, proporcionándoles las herramientas didácticas necesarias para fomentar el pensamiento crítico La Tutoría es un elemento central de la labor docente y en ella el profesor asume plenamente las funciones de facilitador, guía y supervisor de la formación del alumno, tanto en su dimensión científica como personal, para lograr un aprendizaje significativo y las competencias necesarias para su desarrollo profesional. Mediante el instrumento denominado la Carpeta didáctica, el profesor realiza un seguimiento, una dedicación tutorial intensa y personalizada, una recogida de resultados y una evaluación formativa. La Carpeta didáctica es un instrumento de comunicación educativa. Es más que un mero soporte material en cuanto que facilita la organización y estructuración de los nuevos conocimientos. Recoge y pone a disposición del profesor todo lo trabajado por parte del alumno en su secuencia temporal. Ayuda al estudiante en la autorregulación del aprendizaje y en los procesos metacognitivos: aprender a aprender. Incluye la autoevaluación reflexiva y razonada del alumno.The communication that we present is based on an experience of educational innovation carried out carried out with students of the Teaching and Social Education Degrees of the University School of Bilbao Teachers' Association and tries to show to what extent the Autonomous Learning Contract can to be a role model for the teaching-learning of university students. The current demands of university education within the framework of the European Space of Higher Education (EEES) suppose a modification of the modalities and methods of teaching and of the teacher's role. Quality higher education will be that which, beyond transmitting knowledge, introduce meaningful learning situations both inside and outside the classroom, making the student protagonist and responsible for their own learning. The Learning Contract System is based on the constructivist principles of the teaching-learning and highlights the involvement of the student in their training. It is a methodology university that enables the adaptation of the learning objectives to the characteristics and rhythms from the student. It facilitates the autonomous learning of those students who for different reasons cannot Regularly attend classes, providing them with the necessary didactic tools to encourage critical thinking Tutoring is a central element of the teaching work and in it the teacher fully assumes the functions of facilitator, guide and supervisor of the student's training, both in its dimension scientific and personal, to achieve significant learning and the necessary skills for its professional development. Through the instrument called the Didactic Folder, the teacher monitors, an intense and personalized tutorial dedication, a collection of results and a formative evaluation. The Didactic Folder is an educational communication tool. It is more than a mere material support in that it facilitates the organization and structuring of new knowledge. pick up and makes available to the teacher everything worked by the student in their time sequence. Aid the student in the self-regulation of learning and metacognitive processes: learning to learn. It includes the reflective and reasoned self-evaluation of the student.peerReviewe

Combinatorial Koszul Homology, Computations and Applications

A la memoria de mi padre, de quien aprendí que sólo cuando hacemos caso al corazón por encima de la razón, podemos ser felices. A mi madre, que me enseñó que sólo el control de la razón puede llevar a buen puerto los impulsos del corazón. A ambos, de cuya unión he aprendido el necesario equilibrio que hace más sencillo el camino de la vida. A Elena, mi amor, es decir, mi vida. A Juan, Lucía y Míkel, mi vida, es decir, mi amor. To write here the list of all persons to whom I am grateful for their contribution in a direct or indirect manner to the elaboration of this thesis I would need more time, space and memory than I have available. I will only emphasize the names of Luis Javier Hernández Paricio and Werner M. Seiler, my two supervisors, which have given me their advice and support, and have encouraged me and teached me many things during these years, not only in the mathematical aspects of the thesis. I also want to emphasise Julio Rubio, who acted as a “third supervisor ” and has lead me in these first steps int