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. / 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.

Dans cette thèse, nous étudions le comportement d'intégrales oscillantes lorsqu'un paramètre fréquentiel tend vers l'infini. Pour cela, nous considérons la version de la méthode de la phase stationnaire de A. Erdélyi qui couvre le cas d'amplitudes singulières et de phases ayant des points stationnaires d'ordre réel, et qui fournit des estimations explicites de l'erreur. La preuve est entièrement détaillée dans la thèse et la méthode améliorée. De plus nous montrons l'impossibilité de déduire, à partir de cette méthode, des estimations uniformes par rapport à la position du point stationnaire dans le cas d'amplitudes singulières. Afin d'obtenir de telles estimations, nous étendons le lemme de van der Corput au cas d'amplitudes singulières et de points stationnaires d'ordre réel.Ces résultats sont appliqués à des solutions d'équations dispersives sur la droite réelle. La transformée de Fourier de la donnée initiale est à support compact et/ou a un point singulier intégrable. Des développements à un terme et des estimations uniformes dans certains cônes de l'espace-temps sont établis: ceci montre que les paquets d'ondes tendent à être localisés dans certains cônes lorsque le temps tend vers l'infini, décrivant leurs mouvements asymptotiquement en temps.Pour finir, nous considérons des solutions approchées de l'équation de Schrödinger avec potentiel sur la droite réelle, telle que la transformée de Fourier du potentiel est à support compact. En appliquant les méthodes précédentes, nous prouvons que ces solutions approchées tendent à être concentrées dans certains cônes lorsque le temps tend vers l'infini, mettant en évidence des phénomènes de type réflexion et transmission. / In this thesis, we study the asymptotic behaviour of oscillatory integrals for one integration variable with respect to a large parameter. We consider the version of the stationary phase method of A. Erdélyi which covers singular amplitudes and phases with stationary points of real order together with explicit error estimates. The proof, which is only sketched in the original paper, is entirely detailed in the present thesis and the method is improved. Moreover we show the impossibility to derive from this method uniform estimates in the case of singular amplitudes with respect to the position of the stationary point. To obtain such estimates, we extend the classical van der Corput lemma to the case of singular amplitudes and stationary points of real order.These results are then applied to solution formulas of certain dispersive equations on the line, covering Schrödinger-type and hyperbolic examples. We suppose that the Fourier transform of the initial condition is compactly supported and/or has a singular point. Expansions to one term and uniform estimates of the solutions in certain space-time cones are established: this shows that the waves packets tend to be time-asymptotically localized in space-time cones, describing their motions when the time tends to infinity.Finally we consider approximate solutions of the Schrödinger equation on the line with potential, where the Fourier transform of the potential is also supposed to have a compact support. Applying the methods mentioned above, we prove that these approximate solutions tend to be time-asymptotically concentrated in certain space-time cones, exhibiting reflection and transmission type phenomena.

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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 / > / Dissertation (MSc (Applied Mathematics))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted