Academic literature on the topic 'Lemme de Van der Corput'

Les développements actuels de la méthode de Van der Corput pour les sommes d'exponentielles font apparaître la nécessité d'apporter des précisions aux transformations de base A et B. La première partie de cette thèse constitue une étude complète de la transformation B simple; le cas des sommes d'exponentielles avec paramètre est également étudié. Dans la deuxième partie, nous étudions un nouveau procédé de majoration pour les sommes simples d'exponentielles qui consiste à adapter la méthode de Fouvry et Iwaniec à celle de Van der Corput. Les résultats obtenus viennent compléter un tableau de Huxley. Enfin, la troisième partie reprend en détail le lemme de la phase stationnaire, le résultat obtenu donne une estimation (probablement) optimale pour les moyennes d'intégrales oscillantes en vue d'applications à la transformation B simple, double et multiple<br>In modern methods for analytic exponential sums theory, the A and B Van der Corput's process occur in various forms where more accuracy is needed. The' first part of this thesis achieves a complete study of B process for single exponential sums or sums with a parameter. In the second part, Fouvry and Iwaniec's method for multiple exponential sums with monomial is combined with A and B Van der Corput's process to get new bounds for single exponential sums which complete Huxley's table. The third part gives an accurate estimation for single oscillating integrals when the critical point is close to the endpoints of the integration interval which applies to mean values of oscillating integrals such as those that occur in the study of multiple B transform

We present the notions of uniform distribution and discrepancy of sequences contained in the unit interval, as well as an important application of discrepancy in numerical integration by way of the quasi-Monte Carlo method. Some fundamental (and other interesting) results with regards to these notions are presented, along with some detalied and instructive examples and comparisons (some of which not often provided by the literature). We go on to analytical and numerical investigations of the asymptotic behaviour of the discrepancy (in particular for the van der Corput-sequence), and for the general error estimates of the quasi-Monte Carlo method. Using the discoveries from these investigations, we give a conditional proof of the van der Corput theorem. Furthermore, we illustrate that by using low discrepancy sequences (such as the vdC-sequence), a rather fast convergence rate of the quasi-Monte Carlo method may still be achieved, even for situations in which the famous theoretical result, the Koksma inequality, hasbeen rendered unusable.<br>Vi presenterar begreppen likformig distribution och diskrepans hos talföljder på enhetsintervallet, såväl som en viktig tillämpning av diskrepans inom numerisk integration via kvasi-Monte Carlo metoden. Några fundamentala (och andra intressanta) resultat presenteras med avseende på dessa begrepp, tillsammans med några detaljerade och instruktiva exempel och jämförelser (varav några sällan presenterade i litteraturen). Vi går vidare med analytiska och numeriska undersökningar av det asymptotiska beteendet hos diskrepansen (särskilt för van der Corput-följden), såväl som för den allmänna feluppskattningen hos kvasi-Monte Carlo metoden. Utifrån upptäckterna från dessa undersökningar ger vi ett villkorligt bevis av van der Corput's sats, samt illustrerar att man genom att använda lågdiskrepanstalföljder (som van der Corput-följden) fortfarande kan uppnå tämligen snabb konvergenshastighet för kvasi-Monte Carlo metoden. Detta även för situationer där de kända teoretiska resultatet, Koksma's olikhet, är oandvändbart.

Cette thèse porte sur les irrégularités de distribution de suites à une ou plusieurs dimensions et sur leurs applications a l'intégration numérique. Elle comprend trois parties. La première partie est consacrée aux suites unidimensionnelles : estimations de la diaphonie de la suite de Van der Corput à partir de l'étude des sommes exponentielles et étude des suites (n). La deuxième partie porte sur quelques suites classiques en dimension plus grande que une (suites de Fame, suites de Halton). La troisième partie, consacrée aux applications à l'intégration contient de nombreux résultats numériques, permettant de comparer l'efficacité de suites.

The use of Hilbert space theory became an important tool for ergodic theoreticians ever since John von Neumann proved the fundamental Mean Ergodic theorem in Hilbert space. Recurrence is one of the corner stones in the study of dynamical systems. In this dissertation some extended ideas besides those of the basic, well-known recurrence results are investigated. Hilbert space theory proves to be a very useful approach towards the solution of multiple recurrence problems in ergodic theory. Another very important use of Hilbert space theory became evident only relatively recently, when it was realized that non-commutative dynamical systems become accessible to the ergodic theorist through the important Gelfand-Naimark-Segal (GNS) representation of C*-algebras as Hilbert spaces. Through this construction we are enabled to invoke the rich catalogue of Hilbert space ergodic results to approach the more general, and usually more involved, non-commutative extensions of classical ergodic-theoretical results. In order to make this text self-contained, the basic, standard, ergodic-theoretical results are included in this text. In many instances Hilbert space counterparts of these basic results are also stated and proved. Chapters 1 and 2 are devoted to the introduction of these basic ergodic-theoretical results such as an introduction to the idea of measure-theoretic dynamical systems, citing some basic examples, Poincairé’s recurrence, the ergodic theorems of Von Neumann and Birkhoff, ergodicity, mixing and weakly mixing. In Chapter 2 several rudimentary results, which are the basic tools used in proofs, are also given. In Chapter 3 we show how a Hilbert space result, i.e. a variant of a result by Van der Corput for uniformly distributed sequences modulo 1, is used to simplify the proofs of some multiple recurrence problems. First we use it to simplify and clarify the proof of a multiple recurrence result by Furstenberg, and also to extend that result to a more general case, using the same Van der Corput lemma. This may be considered the main result of this thesis, since it supplies an original proof of this result. The Van der Corput lemma helps to simplify many of the tedious terms that are found in Furstenberg’s proof. In Chapter 4 we list and discuss a few important results where classical (commutative) ergodic results were extended to the non-commutative case. As stated before, these extensions are mainly due to the accessibility of Hilbert space theory through the GNS construction. The main result in this section is a result proved by Niculescu, Ströh and Zsidó, which is proved here using a similar Van der Corput lemma as in the commutative case. Although we prove a special case of the theorem by Niculescu, Ströh and Zsidó, the same method (Van der Corput) can be used to prove the generalized result. Copyright 2004, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. Please cite as follows: Beters, FJC 2004, A Hilbert space approach to multiple recurrence in ergodic theory, MSc dissertation, University of Pretoria, Pretoria, viewed yymmdd < http://upetd.up.ac.za/thesis/available/etd-02222006-104936 / ><br>Dissertation (MSc (Applied Mathematics))--University of Pretoria, 2007.<br>Mathematics and Applied Mathematics<br>unrestricted