Spaces of bandlimited functions on compact manifolds
This monograph is structured in four chapters. In Chapter 1, we present the context of our problem and the main results proved in this work. We describe the asymptotic behaviour of the reproducing kernel and the construction of new kernels associated to our spaces with a decay away from the diagonal...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2011 |
| País: | España |
| Institución: | CBUC, CESCA |
| Repositorio: | TDR. Tesis Doctorales en Red |
| OAI Identifier: | oai:www.tdx.cat:10803/123823 |
| Acceso en línea: | http://hdl.handle.net/10803/123823 |
| Access Level: | acceso abierto |
| Palabra clave: | Mostreig (Estadística) Muestreo (Estadística) Sampling (Statistics) Interpolació (Matemàtica) Interpolación Interpolation Funció de banda limitada Función de banda limitada Bandlimited function Espai compacte Espacio compacto Compact space Punts de Fekete Puntos de Fekete Fekete points Ciències Experimentals i Matemàtiques 51 |
| Sumario: | This monograph is structured in four chapters. In Chapter 1, we present the context of our problem and the main results proved in this work. We describe the asymptotic behaviour of the reproducing kernel and the construction of new kernels associated to our spaces with a decay away from the diagonal. We shall also explain some tools that will play a fundamental role in the proof of our results. In Chapter 2, we study the problem of a continuous sampling. The role of a discrete family of sampling is played now by a sequence of sets in the manifold called Logvinenko-Sereda sets. We give a complete geometric characterization. A weaker problem is to find a characterization of the Carleson's measures. This question has been also answered in terms of a geometric condition. In Chapter 3, we provide some (qualitative) necessary and sufficient conditions for interpolation and sampling. We define an analog of the Beurling-Landau's density and prove a quantitative necessary condition for sampling and interpolation following the scheme of Landau in the context of the Paley-Wiener spaces. In Chapter 4, we give an application of the density results obtained in Chapter 3 and study the Fekete arrays on compact manifolds with some restriction. Furthermore, we prove from the results of Chapter 3, the equidistribution of the Fekete families on compact manifolds that have a product property (see Definition 4.1 for more details). The results of this monograph are part of the following articles: - J. Ortega-Cerdà, B. Pridhnani. Carleson measures and Logvinenko-Sereda sets on compact manifolds. Forum Mathematicum, to appear ([OCP11b]). - J. Ortega-Cerdà, B. Pridhnani. Beurling-Landau's density on compact manifolds. Preprint ([OCP11a]). |
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