We present an alternative method of extracting production related differences from time-lapse seismic data sets. Our method is not based on the actual subtraction of the two data sets, risking the enhancement of noise and introduction of artifacts due to local phase rotation and slightly misaligned events. Rather, it mutes events of the monitor survey with respect to the baseline survey based on the magnitudes of coefficients in a sparse and local atomic decomposition. Our technique is demonstrated to be an effective tool for enhancing the time-lapse signal from surveys which have been cross-equalized

Le but de cette thèse est d’étudier l’analyse sur les espaces hpc(Rd,M), la version locale des espaces de Hardy à valeurs opératorielles construits par Tao Mei. Les espaces de Hardy locaux à valeurs opératorielles sont définis par les g-fonctions de Littlewood-Paley tronquées et les fonctions intégrables de Lusin tronquées associées au noyau de Poisson. Nous développons la théorie de Calderón-Zygmund sur hpc(Rd,M); nous étudions la dualité hpcbmocq et l’interpolation. D’après ces résultats, nous obtenons la caractérisation générale de hpc(Rd,M) en remplaçant le noyau de Poisson par des fonctions tests raisonnables. Ceci joue un rôle important dans la décomposition atomique lisse de h1c(Rd,M). En même temps, nous étudions aussi les espaces de Triebel-Lizorkin inhomogènes à valeurs opératorielles Fpα,c(Rd,M). Comme dans le cas classique, ces espaces sont connectés avec des espaces de Hardy locaux à valeurs opératorielles par les potentiels de Bessel. Grâce à l’aide de la théorie de Calderón-Zygmund, nous obtenons les caractérisations de type LittlewoodPaley et de type Lusin par des noyaux plus généraux. Ces caractérisations nous permettent d’étudier différentes propriétés de Fpα,c(Rd,M), en particulier, la décomposition atomique lisse. Ceci est une extension et une amélioration de la décomposition atomique précédente de h1c(Rd,M). Comme une application importante de cette décomposition atomique lisse, nous montrons la bornitude d’opérateurs pseudo-différentiels avec les symboles réguliers à valeurs opératorielles sur des espaces de Triebel-Lizorkin Fpα,c(Rd,M), pour α ∈ R et 1 ≤ p ≤ ∞. Finalement, grâce à la transférence, nous obtenons aussi la Fpα,c-bornitude d’opérateurs pseudo-différentiels sur les tores quantiques / This thesis is devoted to the study of the analysis on the spaces hpc(Rd,M), the local version of operator-valued Hardy spaces studied by Tao Mei. The operator-valued local Hardy spaces are defined by the truncated Littlewood-Paley g-functions and the truncated Lusin square functions associated to the Poisson kernel. We develop the Calderón-Zygmund theory on hpc(Rd,M), and study the hpc-bmocq duality and the interpolation. Based on these results, we obtain general characterization of hpc(Rd,M) which states that the Poisson kernel can be replaced by any reasonable test function. This characterization plays an important role in the smooth atomic decomposition of h1c(Rd,M). We also investigate the operator-valued inhomogeneous Triebel-Lizorkin spaces Fpα,c(Rd,M). Like in the classical case, these spaces are connected with the operator-valued local Hardy spaces via Bessel potentials. Then by the aid of the Calderón-Zygmund theory, we obtain the Littlewood-Paley type and the Lusin type characterizations of Fpα,c(Rd,M) by more general kernels. These characterizations allow us to study various properties of Fpα,c(Rd,M), in particular, the smooth atomic decomposition. This is an extension and an improvement of the previous atomic decomposition of h1c(Rd,M). As an important application of this smooth atomic decomposition, we show the boundedness of pseudo-differential operators with regular operator-valued symbols on Triebel-Lizorkin spaces Fpα,c(Rd,M), for α ∈ R and 1 ≤ p ≤ ∞. Finally, by virtue of transference, we obtain the Fpα,c-boundedness of pseudo-differential operators on quantum tori

Dysfunction in the central nervous system of the neonate is often first identified through seizures. The diffculty in detecting clinical seizures, which involves the observation of physical manifestations characteristic to newborn seizure, has placed greater emphasis on the detection of newborn electroencephalographic (EEG) seizure. The high incidence of newborn seizure has resulted in considerable mortality and morbidity rates in the neonate. Accurate and rapid diagnosis of neonatal seizure is essential for proper treatment and therapy. This has impelled researchers to investigate possible methods for the automatic detection of newborn EEG seizure. This thesis is focused on the development of algorithms for the automatic detection of newborn EEG seizure using adaptive time-frequency signal processing. The assessment of newborn EEG seizure detection algorithms requires large datasets of nonseizure and seizure EEG which are not always readily available and often hard to acquire. This has led to the proposition of realistic models of newborn EEG which can be used to create large datasets for the evaluation and comparison of newborn EEG seizure detection algorithms. In this thesis, we develop two simulation methods which produce synthetic newborn EEG background and seizure. The simulation methods use nonlinear and time-frequency signal processing techniques to allow for the demonstrated nonlinear and nonstationary characteristics of the newborn EEG. Atomic decomposition techniques incorporating redundant time-frequency dictionaries are exciting new signal processing methods which deliver adaptive signal representations or approximations. In this thesis we have investigated two prominent atomic decomposition techniques, matching pursuit and basis pursuit, for their possible use in an automatic seizure detection algorithm. In our investigation, it was shown that matching pursuit generally provided the sparsest (i.e. most compact) approximation for various real and synthetic signals over a wide range of signal approximation levels. For this reason, we chose MP as our preferred atomic decomposition technique for this thesis. A new measure, referred to as structural complexity, which quantifes the level or degree of correlation between signal structures and the decomposition dictionary was proposed. Using the change in structural complexity, a generic method of detecting changes in signal structure was proposed. This detection methodology was then applied to the newborn EEG for the detection of state transition (i.e. nonseizure to seizure state) in the EEG signal. To optimize the seizure detection process, we developed a time-frequency dictionary that is coherent with the newborn EEG seizure state based on the time-frequency analysis of the newborn EEG seizure. It was shown that using the new coherent time-frequency dictionary and the change in structural complexity, we can detect the transition from nonseizure to seizure states in synthetic and real newborn EEG. Repetitive spiking in the EEG is a classic feature of newborn EEG seizure. Therefore, the automatic detection of spikes can be fundamental in the detection of newborn EEG seizure. The capacity of two adaptive time-frequency signal processing techniques to detect spikes was investigated. It was shown that a relationship between the EEG epoch length and the number of repetitive spikes governs the ability of both matching pursuit and adaptive spectrogram in detecting repetitive spikes. However, it was demonstrated that the law was less restrictive forth eadaptive spectrogram and it was shown to outperform matching pursuit in detecting repetitive spikes. The method of adapting the window length associated with the adaptive spectrogram used in this thesis was the maximum correlation criterion. It was observed that for the time instants where signal spikes occurred, the optimal window lengths selected by the maximum correlation criterion were small. Therefore, spike detection directly from the adaptive window optimization method was demonstrated and also shown to outperform matching pursuit. An automatic newborn EEG seizure detection algorithm was proposed based on the detection of repetitive spikes using the adaptive window optimization method. The algorithm shows excellent performance with real EEG data. A comparison of the proposed algorithm with four well documented newborn EEG seizure detection algorithms is provided. The results of the comparison show that the proposed algorithm has significantly better performance than the existing algorithms (i.e. Our proposed algorithm achieved a good detection rate (GDR) of 94% and false detection rate (FDR) of 2.3% compared with the leading algorithm which only produced a GDR of 62% and FDR of 16%). In summary, the novel contribution of this thesis to the fields of time-frequency signal processing and biomedical engineering is the successful development and application of sophisticated algorithms based on adaptive time-frequency signal processing techniques to the solution of automatic newborn EEG seizure detection.

Made available in DSpace on 2016-06-02T20:28:30Z (GMT). No. of bitstreams: 1 6065.pdf: 865751 bytes, checksum: 22466d8659637f2282b6be8b0adb5a33 (MD5) Previous issue date: 2014-03-13 / Financiadora de Estudos e Projetos / This work is divided into two parts. In the first part, our goal is to present the theory of Hardy Spaces Hp(Rn), which coincides with the Lebesgue space Lp(Rn) for p > 1, is strictly contained in Lp(Rn) if p = 1, and is a space of distributions when 0 < p < 1. When 0 < p ^ 1, the Hardy spaces offers a better treatment involving harmonic analysis than the Lp spaces. Among other results, we prove the maximal characterization theorem of Hp, which gives equivalent definitions of Hp, based on different maximal functions. We will proof the atomic decom¬position theorem for Hp, which allow decompose any distribution in Hp to be written as a sum of Hp-atoms (measurable functions that satisfy certain properties). In this step, we use the strongly the of Whitney decomposition and generalized Calderon-Zygmund decomposition. In the second part, as a application, we will prove that nonlinear quantities (such as the Jacobian, divergent and rotational defined in Rn) identied by the compensated compactness theory belong, under natural conditions, the Hardy spaces. To this end, in addition to the results seen in the first part, will use the results as Sobolev immersions theorems ans the inequality Sobolev-Poincare. Furthermore, we will use the tings and results related to the context of differential forms. / Esse trabalho está dividido em duas partes.Na primeira, nosso objetivo e apresentar os espaços de Hardy Hp(Rn), o qual coincide com os espaços Lp(Rn), quando p > 1, esta estritamente contido em Lp(Rn) se p = 1, e e um espaço de distribuições quando 0 < p < 1. Quando 0 < p < 1, os espaços de Hardy oferecem um melhor tratamento envolvendo analise harmônica do que os espaços Lp(Rn). Entre outros resultados, provamos o teorema da caracterização maximal de Hp, o qual fornece varias, porem equivalentes, formas de caracterizar Hp, com base em diferentes funcões maximais. Demonstramos o teorema da decomposição atômica para Hp, 0 < p < 1, que permite decompor qualquer distribuição em Hp como soma de Hp-atomos (funções mensuráveis que satisfazem certas propriedades). Nessa etapa, usamos fortemente a de- composição de Whitney e a decomposto de Calderon-Zygmund generalizada. Na segunda parte, como uma aplicação, provamos que quantidades não-lineares (como o jacobiano, divergente e o rotacional definidos em Rn), identificadas pela teoria compacidade compensada pertencem, em condições naturais, aos espaços de Hardy. Para tanto, além dos resultados visto na primeira parte, usamos outros como os Teoremas de Imersões de Sobolev, a desigualdade de Sobolev-Poincaré. Usamos ainda, definições e resultados referentes ao contexto de formas diferenciais.

[EN] The Ph.D. Thesis "Atomic decompositions and frames in Fréchet spaces and their duals" presented here treats different areas of functional analysis with applications. Schauder frames are used to represent an arbitrary element x of a function space E as a series expansion involving a fixed countable set {xj} of elements in that space such that the coefficients of the expansion of x depend in a linear and continuous way on x. Unlike Schauder bases, the expression of an element x in terms of the sequence {xj}, i.e. the reconstruction formula for x, is not necessarily unique. Atomic decompositions or Schauder frames are a less restrictive structure than bases, because a complemented subspace of a Banach space with basis has always a natural Schauder frame, that is obtained from the basis of the superspace. Even when the complemented subspace has a basis, there is not a systematic way to find it. Atomic decompositions appeared in applications to signal processing and sampling theory among other areas. Very recently, Pilipovic and Stoeva [55] studied series expansions in (countable) projective or inductive limits of Banach spaces. In this thesis we begin a systematic study of Schauder frames in locally convex spaces, but our main interest lies in Fréchet spaces and their duals. The main difference with respect to the concept considered in [55] is that our approach does not depend on a fixed representation of the Fréchet space as a projective limit of Banach spaces. The text is divided into two chapters and appendix that gives the notation, definitions and the basic results we will use throughout the thesis. The first one focuses on the relation between the properties of an existing Schauder frame in a Fréchet space E and the structure of the space. In the second chapter frames and Bessel sequences in Fréchet spaces and their duals are defined and studied. In what follows, we give a brief description of the different chapters: In Chapter 1, we study Schauder frames in Fréchet spaces and their duals, as well as perturbation results. We define shrinking and boundedly complete Schauder xviiframes on a locally convex space, study the duality of these two concepts and their relation with the reflexivity of the space. We characterize when an unconditional Schauder frame is shrinking or boundedly complete in terms of properties of the space. Several examples of concrete Schauder frames in function spaces are also presented. Most of the results included in this chapter are published by Bonet, Fernández, Galbis and Ribera in [13]. The second chapter of the thesis is devoted to study ¿-Bessel sequences, ¿-frames and frames with respect to ¿ in the dual of a Hausdorff locally convex space E, in particular for Fréchet spaces and complete (LB)-spaces E, with ¿ a sequence space. We investigate the relation of these concepts with representing systems in the sense of Kadets and Korobeinik [34] and with the Schauder frames, that were investigated in Chapter 1. The abstract results presented here, when applied to concrete spaces of analytic functions, give many examples and consequences about sampling sets and Dirichlet series expansions. We present several abstract results about ¿-frames in complete (LB)-spaces. Finally, many applications, results and examples concerning sufficient sets for weighted Fréchet spaces of holomorphic functions and weakly sufficient sets for weighted (LB)-spaces of holomorphic functions are collected. Most of the results are submitted for publication in a preprint of Bonet, Fernández, Galbis and Ribera in [12]. / [ES] La presente memoria "Descomposiciones atómicas y frames en espacios de Fréchet y sus duales" trata diferentes áreas del análisis funcional con aplicaciones. Los frames de Schauder se utilizan para representar un elemento arbitrario x de un espacio de funciones E mediante una serie a partir de un conjunto numerable fijado {xj} de elementos de este espacio de manera que los coeficientes de la reconstrucción de x dependen de forma lineal y continua de x. A diferencia de las bases de Schauder, la expresión de un elemento x en términos de la sucesión {xj}, i.e. la fórmula de reconstrucción para x, no es necesariamente única. Las descomposiciones atómicas o los frames de Schauder son un estructura menos restrictiva que las bases, porque un subespacio complementado de un espacio de Banach con base tiene siempre un frame de Schauder natural, que se obtiene a partir de una base del superespacio. Incluso cuando el subespacio complementado tiene una base, no hay una forma sistemática de encontrarla. Las descomposiciones atómicas aparecen en aplicaciones al procesamiento de señales y la teoría de muestreo, entre otras áreas. Recientemente, Pilipovic y Stoeva [55] han estudiado el desarrollo en serie en límites inductivos y proyectivos (numerables) de espacios de Banach. En esta tesis empezamos un estudio sistemático de los frames de Schauder en espacios localmente convexos aunque nuestro interés principal son los espacios de Fréchet y sus duales. La diferencia principal respecto del concepto considerado en [55] es que nuestra aproximación no depende de una representación fijada del espacio de Fréchet como límite proyectivo de espacios de Banach. El texto queda dividido en dos partes y un apéndice que incluye la notación, las definiciones y los resultados básicos que usaremos a lo largo de la tesis. La primera parte se centra en la relación entre las propiedades de un frame de Schauder en un espacio de Fréchet E y la estructura del espacio. En el segundo capítulo se definen y estudian los frames y las sucesiones de Bessel en espacios de Fréchet y sus duales. A continuación, presentamos una breve descripción de los capítulos: En el Capítulo 1, estudiamos los frames de Schauder en los espacios de Fréchet y sus duales así como los resultados de perturbación. Definimos los frames de Schauder contractivos y acotadamente completos en espacios localmente convexos, estudiamos la dualidad de estos dos conceptos y su relación con la reflexividad del espacio. Caracterizamos cuándo un frame de Schauder incondicional es contractivo o acotadamente completo en términos de las propiedades del espacio. También se presentan varios ejemplos de frames de Schauder en espacios de funciones concretos. La mayoría de los resultados incluidos en este capítulo están publicados por Bonet, Fernández, Galbis y Ribera en [13]. El segundo capítulo de la tesis está centrado en el estudio de las sucesiones de ¿-Bessel, ¿-frames y frames respecto de ¿ en el dual de un espacio localmente convexo de Hausdorff E, en particular, para espacios de Fréchet y espacios (LB) completos E, con ¿ un espacio de sucesiones. Investigamos la relación de estos dos conceptos con los sistemas representantes en el sentido de Kadets y Korobeinik [34] y con los frames de Schauder, considerados en el Capítulo 1. Los resultados abstractos presentados aquí, cuando los aplicamos a espacios de funciones analíticas concretos, nos dan muchos ejemplos y consecuencias sobre los conjuntos de muestreo y los desarrollos en serie de Dirichlet. Presentamos varios resultados abstractos sobre ¿-frames en espacios (LB) completos. Finalmente, recogemos muchas aplicaciones, resultados y ejemplos alrededor de los conjuntos suficientes para espacios de Fréchet de funciones holomorfas y conjuntos débilmente suficientes para espacios pesados (LB) de funciones holomorfas. La mayoría de los resultados incluidos en este capítulo están enviados para publicar e / [CA] La tesi "Descomposicions atòmiques i frames en espais de Fréchet i els seus duals" presentada ací tracta diferents àrees de l'anàlisi funcional amb aplicacions. Els frames de Schauder s'utilitzen per tal de representar un element arbitrari x d'un espai de funcions E com una reconstrucció en sèrie a partir d'un conjunt numerable fixat {xj} d'elements en aquest espai tal que els coeficients de la reconstrucció de x depenen de forma lineal i continua de x. A diferència de les bases de Schauder, l'expressió d'un element x en termes d'una successió {xj}, i.e. la fórmula de reconstrucció per a x, no és necessàriament única. Les descomposicions atòmiques o els frames de Schauder són una estructura menys restrictiva que les bases, donat que un subespai complementat d'un espai de Banach amb base sempre té un frame de Schauder natural, el qual és obtingut a partir d'una base del superespai. Inclòs quan el subespai complementat disposa de una base, no hi ha una forma sistemàtica per tal de trobar-la. Les descomposicions atòmiques apareixen en aplicacions a processat de senyals i teoría de mostreig entre altres àrees. Recentment, Pilipovic i Stoeva [55] han estudiat els desenvolupaments en sèrie en límits inductius o projectius (numerables) en espais de Banach. En aquesta tesi comencem un estudi sistemàtic dels frames de Schauder en espais localment convexos, tot i que el nostre interés està en els espais de Fréchet i els seus duals. La diferència més important amb el concepte estudiat en [55] és que el nostre estudi no depén de una representació fixada del espai de Fréchet com a límit projectiu de espais de Banach. El text està dividit en dos capítols i un apèndix que ens aporta la notació, definicions i els resultats bàsics que utilitzarem al llarg de la tesi. El primer dels capítols està centrat en la relació entre les propietats de un frame de Schauder en un espai de Fréchet E i la estructura del espai. En el segon capítol es defineixen i estudien els frames i les successions de Bessel en espais de Fréchet i els seus duals. En el que segueix, donem una breu descripció dels diferents capítols: En el Capítol 1, estudiem els frames de Schauder en els espais de Fréchet i els seus duals, així com els resultats de pertorbació. Definim els frames de Schauder contractius i fitadament complets en espais localment convexos, estudiem la dualitat d'aquests dos conceptes i la seua relació amb la reflexivitat del espai. Caracteritzem, en quines situacions, un frame de Schauder incondicional és contractiu o fitadament complet en termes de les propietats del espai. També presentem alguns exemples de frames de Schauder concrets en espais de funcions. La majoria dels resultats inclosos en aquest capítol estan publicats per Bonet, Fernández, Galbis i Ribera en [13]. El segon capítol de la tesi està centrat en el estudi de les successions ¿-Bessel, ¿-frames i frames respecte de ¿ en el dual d'un espai localment convex de Hausdorff E, en particular, per a espais de Fréchet i espais (LB) complets E, amb ¿ un espai de successions. Investiguem la relació d'aquests dos conceptes amb sistemes representants en el sentit de Kadets i Korobeinik [34] i amb els frames de Schauder, que han sigut investigats en el Capítol 1. Els resultats abstractes presentats ací, quan els apliquem a espais de funcions analítiques concrets, ens donen molts exemples i conseqüències sobre els conjunts de mostreig i els desenvolupaments en sèrie de Dirichlet. Presentem diversos resultats abstractes sobre ¿-frames en espais (LB) complets. Finalment, recollim moltes aplicacions, resultats i exemples al voltant dels conjunts suficients per a espais de Fréchet de funcions holomorfes i conjunts dèbilment suficients per a espais pesats (LB) de funcions holomorfes. La majoria dels resultats inclosos en aquest capítol estan sotmesos a publicació per Bonet, Fernández, Galbis i Ribera en [12]. / Ribera Puchades, JM. (2015). Atomic decompositions and frames in Fréchet spaces and their duals [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/49987

Cette thèse présente quelques résultats de la théorie des probabilités quantiques et de l’analyse harmonique à valeurs operateurs. La thèse est composée des trois parties.Dans la première partie, on démontre la décomposition atomique des espaces de Hardy de martingales non commutatives. On identifie aussi les interpolés complexes et réels entre les versions conditionnelles des espaces de Hardy et BMO de martingales non commutatives.La seconde partie est consacrée à l’étude des espaces de Hardy à valeurs opérateursvia la méthode d’ondellettes. Cette approche est similaire à celle du cas des martingales non commutatives. On démontre que ces espaces de Hardy sont équivalents à ceux étudiés par Tao Mei. Par conséquent, on donne une base explicite complètement inconditionnelle pour l’espace de Hardy H1(R), muni d’une structure d’espace d’opérateurs naturelle. La troisième partie porte sur l’analyse harmonique sur le tore quantique. On établit les inégalités maximales pour diverses moyennes de sommation des séries de Fourier définies sur le tore quantique et obtient les théorèmes de convergence ponctuelle correspondant. En particulier, on obtient un analogue non commutative du théorème classique de Stein sur les moyennes de Bochner-Riesz. Ensuite, on démontre que les multiplicateurs de Fourier complètement bornés sur le tore quantique coïncident à ceux définis sur le tore classique. Finalement, on présente la théorie des espaces de Hardy et montre que ces espaces possèdent les propriétés des espaces de Hardy usuels. En particulier, on établit la dualité entre H1 et BMO. / This thesis presents some results in quantum probability and operator-valued harmonicanalysis. The main results obtained in the thesis are contained in the following three parts:In first part, we prove the atomic decomposition for the Hardy spaces h1 and H1 of noncommutative martingales. We also establish that interpolation results on the conditionedHardy spaces of noncommutative martingales. The second part is devoted to studying operator-valued Hardy spaces via Meyer’s wavelet method. It turns out that this way of approaching these spaces is parallel to that in the noncommutative martingale case. We also show that these Hardy spaces coincide with those introduced and studied by Tao Mei in [52]. As a consequence, we give an explicit completely unconditional base for Hardy spaces H1(R) equipped with a natural operator space structure. The third part deals with with harmonic analysis on quantum tori. We first establish the maximal inequalities for several means of Fourier series defined on quantum tori and obtain the corresponding pointwise convergence theorems. In particular, we prove the noncommutative analogue of the classical Stein theorem on Bochner-Riesz means. Then we prove that Lp completely bounded Fourier multipliers on quantum tori coincide with those on classical tori with equal cb-norms. Finally, we present the H1-BMO and Littlewood- Paley theories associated with the circular Poisson semigroup over quantum tori.

Quelques problèmes en analyse harmonique non commutative / Some problems on noncommutative harmonique analysis