In this paper I formulate an explicit wavelet transform that, applied to any distribution in S^1(R^2), yields a function on phase space whose high-frequency singularities coincide precisely with the wave front set of the distribution. This characterizes the wave front set of a distribution in terms of the singularities of its wavelet transform with respect to a suitably chosen basic wavelet.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-06-26T17:28:14Z No. of bitstreams: 1 joseamanciodossantos.pdf: 408905 bytes, checksum: 5a4696372063642f4350d6dbd066da13 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-08-07T20:37:22Z (GMT) No. of bitstreams: 1 joseamanciodossantos.pdf: 408905 bytes, checksum: 5a4696372063642f4350d6dbd066da13 (MD5) / Made available in DSpace on 2017-08-07T20:37:22Z (GMT). No. of bitstreams: 1 joseamanciodossantos.pdf: 408905 bytes, checksum: 5a4696372063642f4350d6dbd066da13 (MD5) Previous issue date: 2010-06-16 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Como é bem conhecido, os campos quânticos estudados na TQC satisfazem o princípio de localidade segundo pontos do espaço-tempo. Entretanto, os príncipios da física também admitem campos quânticos que satisfazem a uma condição de localidade determinada por strings 1, as quais são semi-retas no espaço-tempo partindo de algum ponto (evento) e se estendendo em alguma direção do tipo espaço. Devido a esta noção de localidade via string, dizemos que tais campos possuem localização do tipo-string. Por outro lado, nos referimos aos campos cuja localidade é caracterizada por pontos do espaço-tempo, dizendo que eles possuem localização do tipo-ponto ou que são puntiformemente localizados. O interesse na localização do tipo-string está na possibilidade de campos com tal localização apresentarem um comportamento UV, isto é, em altas energias, menos singular do que os campos com a localização do tipo-ponto, permitindo assim a obtenção de mais modelos interagentes com localização do tipo-string. Campos livres com localização tipo-string já foram obtidos para vários tipos de partículas [1, 2], a partir dos quais pode-se construir modelos interagentes. No entanto, para realizar esta tarefa, ou seja, construir modelos com interação a partir do campo livre, deve-se fazer uma análise da função de dois pontos do campo livre correspondente. Neste ponto se faz necessário o uso de certos conceitos e instrumentos - por exemplo: suporte singular, wave front set e scaling degree - na análise da função de dois pontos. Neste texto procuramos introduzir estes conceitos e instrumentos. Além disso, consideramos um modelo de campo escalar livre com localização do tipo-string para uma partícula massiva com spin nulo, para o qual apresentamos e procuramos analisar a estrutura de singularidades da função de dois pontos correspondente, dando uma interpretação em termos de strings. / As is well-known, the quantum fields studied in QFT satisfy the principle of locality according to points in space-time. However, the principles of physics also admit quantum fields that satisfy a condition of locality determined by strings2, which are rays (semi axes) in space-time starting from some point (event) and extending in some space-like direction. Due to this notion of locality via string, we say that such fields are string-localized. On the other hand, we refer to fields whose locality is characterized by points in space-time, saying that they are localized on points. The interest in string localization is the possibility that fields with such kind of localization present a less singular UV behaviour, that is, at high energy, than that of fields localized on points, and then permitting the construction of more interacting models. String-localized free quantum fields have been constructed for many particles types [1, 2], from which one can construct interacting models. However, in order to do this, that is, to construct interacting models from the free fields, it is necessary to analyse the two point function of the corresponding free fields. At this point we have to use some concepts and tools - for example: singular support, wave front set and scaling degree - to analyse the two point function. In this text we introduce these concepts and tools. Moreover, we consider a string-localized free scalar quantum field model for a massive spin zero particle, for which we present and analyse the singularity structure of the corresponding two point function, giving a interpretation in terms of strings.

<p>We introduce a family of smooth functions which are &quot;less regu-lar&quot; than the Gevrey functions, and study its basic properties. In particular&nbsp;we prove the standard results concerning algebra property and stability under finite order derivation. Moreover, we &nbsp;construct infnite order operators&nbsp;which leads us to the definition of class with ultradifferentiable property. We&nbsp;also prove that our classes are inverse-closed, and this result is the essential&nbsp;part in the proof of our main result presented in the final Chapter. Moreover,&nbsp;using the techniques of microlocal analysis, we introduce and investigate the<br />corresponding wave front sets, and the prove the results related to singular&nbsp;support of a distribution. Our main results shows how the singularities of&nbsp;solutions to partial differential equations (PDE&#39;s in short) propagate in the&nbsp;framework of our regularity.</p> / <p>U ovoj tezi defini&scaron;emo novu klasu glatkih funkcija i izučavamo njihove osnovne osobine. Pokazujemo da na&scaron;e klase imaju svojsto algebre kao i da su zatvorene u odnosu na delovanje operatora izvoda konačnog reda.Sta vi&scaron;e, konstrui&scaron;emo diferencijalne operatore beskonačnog reda i to nas dovodi do definicije ultradiferencijabilnih klasa funkcija. Takode dokazujemo osobinu zatvorenosti u odnosu na inverze, i taj rezultat je najvažniji deo u dokazu glavne teoreme koja je formulisana u poslednjoj glavi. Koristeći tehnike mikrolokalne analize, uvodimo i izučavamo odgovarajuće talasne frontove, i pokazujemo odgovarajuća tvrdjenja vezana za singularni nosač distribucije. Na&scaron; glavni rezultat pokazuje kako se prostiru singulariteti re&scaron;enja linearnih parcijalnih diferencijalnih jednačina u okviru na&scaron;e regularnosti.</p>

Diese Dissertation ist dem Studium temperierter Distributionen mittels mikrolokaler Methoden gewidmet. Die fundamentale Größe der mikrolokalen Analysis, die Wellenfrontmenge, wird durch zwei analoge Konzepte ersetzt, die den pseudo-differentiellen SG- und Shubin-Kalkülen zugeordnet sind. Die Eigenschaften dieser globalen Wellenfrontmengen werden studiert und ferner werden unterschiedliche Möglichkeiten, diese globalen Singularitäten zu charakterisieren, untersucht, insbesondere mittels der FBI-Transformation. Zahlreiche Konstruktionen, die den klassischen Wellenfrontmengenbegriff beinhalten, werden in den globalen Kontext übersetzt, insbesondere Rechenoperationen mit temperierten Distributionen wie etwa (getwistete) Produkte, Pull-backs und Paarungen, für die mikrolokale Existenzkriterien angegeben werden. Als eine Anwendung wird eine Klasse von temperierten Oszillatorintegralen eingeführt, welche durch inhomogene Phasenfunktionen und Amplituden aus SG-Symbolklassen parametrisiert werden. Die SG-Wellenfrontmengen dieser Distributionen werden untersucht und es stellt sich heraus, dass diese durch eine Verallgemeinerung der Menge stationärer Punkte der Phasenfunktionen beschränkt werden. In diesem Kontext wird eine Verallgemeinerung des klassischen Begriffs einer konischen Lagrange-Untermannifaltigkeit des T*R^d vorgenommen und diese Objekte werden auf ihre Parametrisierungseigenschaften untersucht. Es stellt sich heraus, dass jedes solche Objekt lokal als die Menge der stationären Punkte einer SG-Phasenfunktion realisiert werden kann. Als weitere Anwendung werden einige Konstruktionen der axiomatischen Quantenfeldtheorie, die Distributionen beinhalten, im temperierten Kontext realisiert.

[ES] En esta tesis estudiamos operadores pseudodiferenciales, que son operadores integrales de la forma f 7→ ∫ Rd (∫ Rd ei(x−y)·ξa(x,y,ξ)f(y)dy)dξ, en las clases globales de funciones ultradiferenciables de tipo Beurling Sω(Rd) introducidas por Björck, cuando la función peso ω viene dada en el sentido de Braun, Meise y Taylor. Desarrollamos el cálculo simbólico para estos operadores, tratando además el cambio de cuantización, la existencia de paramétrix pseudodiferencial y aplicaciones al frente de ondas global. La tesis consta de cuatro capítulos. En el Capítulo 1 introducimos los símbolos y amplitudes globales, y demostramos que los correspondientes operadores pseudodiferenciales están bien definidos y son continuos en en Sω(Rd). Estos resultados son extendidos en el Capítulo 2 para cuantizaciones arbitrarias, lo que conduce al estudio del traspuesto de cualquier cuantización de un operador pseudodiferencial y a la composición de dos cuantizaciones distintas de operadores pseudodiferenciales. En el Capítulo 3, desarrollamos el método de la paramétrix, dando condiciones suficientes para la existencia de paramétrix por la izquierda de un operador pseudodiferencial, que motiva en el Capítulo 4 la definición de un nuevo frente de ondas global para ultradistribuciones en S′ω(Rd) dada en términos de cuantizaciones de Weyl. Comparamos este frente de ondas con el frente de ondas de Gabor definido mediante la STFT y damos aplicaciones a la regularidad de las cuantizaciones de Weyl. / [CAT] En aquesta tesi estudiem operadors pseudodiferencials, que són operadors integrals de la forma f 7→ ∫ Rd (∫ Rd ei(x−y)·ξa(x,y,ξ)f(y)dy)dξ, en les classes globals de funcions ultradiferenciables de tipus Beurling Sω(Rd) introduïdes per Björck, quan la funció pes ω ve donada en el sentit de Braun, Meise i Taylor. Desenvolupem el càlcul simbòlic per aquestos operadors, tractant, a més a més, el canvi de quantització, l'existència de paramètrix pseudodiferencial i aplicacions al front d'ones global. La tesi consisteix de quatre capítols. Al Capítol 1 introduïm els símbols i amplituds globals, i demostrem que els corresponents operadors pseudodiferencials estan ben definits i són continus en Sω(Rd). Aquestos resultats són estesos al Capítol 2 per a quantitzacions arbitràries, que condueix a l'estudi del transposat de qualsevol quantització d'un operador pseudodiferencial i a la composició de dues quantitzacions distintes d'operadors pseudodiferencials. Al Capítol 3 desenvolupem el mètode de la paramètrix, donant condicions suficients per a l'existència de paramètrix per l'esquerra d'un operador pseudodiferencial donat, que motiva al Capítol 4 la definició d'un nou front d'ones global per a ultradistribucions en S′ω(Rd) mitjançant quantitzacions de Weyl. Comparem aquest front d'ones amb el front d'ones de Gabor definit mitjançant la STFT i donem aplicacions a la regularitat de les quantitzacions de Weyl. / [EN] In this thesis we study pseudodifferential operators, which are integral operators of the form f 7→ ∫ Rd (∫ Rd ei(x−y)·ξa(x,y,ξ)f(y)dy)dξ, in the global class of ultradifferentiable functions of Beurling type Sω(Rd) as introduced by Björck, when the weight function ω is given in the sense of Braun, Meise, and Taylor. We develop a symbolic calculus for these operators, treating also the change of quantization, the existence of pseudodifferential parametrices and applications to global wave front sets. The thesis consists of four chapters. In Chapter 1 we introduce global symbols and amplitudes and show that the corresponding pseudodifferential operators are well defined and continuous in Sω(Rd). These results are extended in Chapter 2 for arbitrary quantizations, which leads to the study of the transpose of any quantization of a pseudodifferential operator, and the composition of two different quantizations of pseudodifferential operators. In Chapter 3 we develop the method of the parametrix, providing sufficient conditions for the existence of left parametrices of a pseudodifferential operator, which motivates in Chapter 4 the definition of a new global wave front set for ultradistributions in S′ω(Rd) given in terms of Weyl quantizations. Then, we compare this wave front set with the Gabor wave front set defined by the STFT and give applications to the regularity of Weyl quantizations. / Asensio López, V. (2021). Global pseudodifferential operators in spaces of ultradifferentiable functions [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/174847 / TESIS