<p>We investigate the Laplace transform in Komatsu ultradistributions and give conditions under which an analytic function is a Laplace transformation of an ultradistribution. We&nbsp; prove the equivalence of several denitions of convolu-tion of two Roumieu ultradistributions. For that purpose, we consider the _ ten-sor product of _~BfMpg<br />and a locally convex space. We dene specic global symbol classes of Shubin type and study the corresponding pseudodierential operators of innite order that act continuously on the spaces of tempered ultradistributions of Beurling and Roumieu type. For these classes, we develop symbolic calculus. We investigate the connection between the Anti-Wick and Weyl quantization when the symbols belong to these classes. We nd the largest subspace of ultradistri-butions for which the convolution with the gaussian kernel exists. This gives a way to extend the denition of Anti-Wick quantization for symbols that are not necessarily tempered ultradistributions.</p> / <p>Prouqavamo Laplasovu transformaciju u prostorima Komat-suove ultradistribucije i dajemo uslov pod kojim analitiqka funk-cija je Laplasova transformacija ultradistribucije. Dokazujemo ek-vivalentnost nekoliko definicija o konvoluciji dve Rumie ultradis-tribucije. Za &nbsp; ovu svrhu razmatramo &quot; tenzorski proizvod&nbsp; ~ B fMpg i lokalno konveksni prostor. Definiramo specifiqne globalne simbol klase Xubinovog tipa i prouqavamo odgovarajue psevdo diferenci-jalne operatore beskonaqnog reda koji neprekidno deluju na prosto-rima temperiranih ultradistribucija Berlineovog i Rumieovog tipa. Za ove klase gradimo simboliqki&nbsp; kalkulus. Prouqavamo vezu izmeu Anti-Wick-ove i Weyl-ove kvantizacije kad simboli pripadaju ove sim-bol klase. Nalazimo najvei podprostor ultradistribucija za koje konvolucija sa gausovog jezgra postoji. To prua mogunost da pro-xirimo definiciju Anti-Wick kvantizacije za simbole koje nemoraju da su temperirane ultradistribucije.</p>

In this thesis we study the spaces of Gevrey functions and ultradistributions, focusing primarily on the properties reflected on their Fourier-Laplace transforms. In particular, we study the Paley-Wiener Theorems for compactly supported Gevrey functions and compactly supported Gevrey ultradistributions.

<p>U disertaciji se navode definicije i svojstva malotalasne transformacije i kvaziasimptotskog<br />ponaˇsanja distribucija. Teoreme Abelovog i Tauberovog tipa<br />su koriˇs&acute;cene za asimptotsku analizu temperiranih distribucija, u odnosu na<br />njihovu malotalasnu transformaciju. Takode, prouˇcavana je i malotalasna<br />transformacija ultradiferencijabilnih funkcija i temperiranih ultradistribucija.</p> / <p> In thesis we give basic notions of wavelet transform and quasiasymptotic<br /> behavior of distributions. Via Abelian and Tauberian type of theorems we<br /> study quasiasymptotic behavior of tempered distributions, related to their<br /> wavelet transform. Further, we study wavelet transform of ultradifferential<br /> functions and tempered ultradistributions.</p>

<p>We use common notation &lowast; for distribution (Scshwartz), (M_p) (Beurling) i {M_p} (Roumieu) setting. We introduce and study new (ultra) distribution spaces, the test function spaces&nbsp;<em>D^&lowast;_E</em>&nbsp; and their strong duals <em>D^{<span style="font-size: 10px;">&#39;</span>&lowast;}_E&rsquo;*</em>.These spaces generalize the spaces <em>D^&lowast;_L^q , D&#39;^&lowast;_L^p , B&rsquo;*</em>&nbsp;and their weighted versions. The construction of our new (ultra)distribution &nbsp;spaces is based on the analysis of a suitable translation-invariant Banach space of (ultra)distribution <em>E</em>&nbsp;with continuous translation group, which turns out to be a convolution module over the Beurling algebra&nbsp;<em>L¹_&omega;</em>, where the weight &nbsp;&omega; is related to the translation operators on <em>E</em>.&nbsp;The&nbsp;Banach space&nbsp;<em>E</em>^&rsquo;_&lowast;&nbsp;stands for&nbsp;<em>L¹_&omega;ˇ &lowast; E</em>&rsquo;.&nbsp;We apply our results to the study of the&nbsp;convolution of ultradistributions. The spaces of convolutors&nbsp;<em>O<span style="font-size: 12px;">&rsquo;^&lowast;</span><span style="font-size: 8.33333px;">C</span></em><span style="font-size: 12px;"><em>&nbsp;(</em><strong>R</strong><em>ⁿ)</em>&nbsp;</span>for tempered&nbsp;ultradistributions are analyzed via the duality with respect to the test function<br />spaces<span style="font-size: 12px;">&nbsp;<em>O^&lowast;_C (</em><strong>R</strong><em>ⁿ)</em>,&nbsp;</span>introduced in this thesis. Using the properties of translationinvariant<br />Banach space of ultradistributions <em>E</em> we obtain a full characterization of<br />the general convolution of Roumieu ultradistributions via the space of integrable<br />ultradistributions is obtained. We show: The convolution of two Roumieu ultradistributions&nbsp;<span style="font-size: 12px;"><em>T, S &isin; D&rsquo;^{Mp} (</em><strong>R</strong><em>ⁿ)&nbsp;</em> exists if and only if&nbsp;<em>(</em></span><em>&phi;</em><span style="font-size: 12px;"><em>&nbsp;&lowast; &Scaron;) T &isin; D^&rsquo;{Mp}_L¹(</em><strong>R</strong><em>ⁿ)</em>&nbsp; for every </span><em>&phi;</em><span style="font-size: 12px;"><em>&nbsp;&isin; D ^{Mp} (</em><strong>R</strong><em>ⁿ)</em>.&nbsp;</span>We study boundary values of holomorphic functions defined in tube domains. New edge of the wedge theorems are obtained. The results<br />are then applied to represent<span style="font-size: 12px;">&nbsp;<em>D&rsquo;_E&rsquo;*</em></span><span style="font-size: 12px;">&nbsp;&nbsp;</span>as a quotient space of holomorphic functions.<br />We also give representations of elements of<span style="font-size: 12px;">&nbsp;<em>D&rsquo;_E&rsquo;*</em></span><span style="font-size: 12px;">&nbsp;&nbsp;</span>via the heat kernel method.</p> / <p>Koristimo oznaku &lowast; za distribuciono (Svarcovo), (Mp) (Berlingovo) i&nbsp;{Mp} (Roumieuovo) okruženje. Uvodimo i prouavamo nove (ultra)distribucione&nbsp;prostore, &nbsp;test funkcijske prostore <em>D</em>^&lowast;_E i njihove duale <em>D^&#39;</em>^&lowast;_{<em>E&#39;*</em>}.&nbsp;&nbsp;Ovi prostori uop&scaron;tavaju&nbsp;<br />prostore <em>D</em>^&lowast;_Lq , <em>D</em>^{&#39;&lowast;}_Lp , <em>B^&#39;</em>^&lowast; i njihove težinske verzije. Konstrukcija na&scaron;ih novih&nbsp;<br />(ultra)distribucionih prostora je zasnovana na analizi odgovarajuićh translaciono&nbsp;<br />- invarijantnih Banahovih prostora (ultra)distribucija koje označavamo sa&nbsp;<em>E</em>. Ovi prostori imaju neprekidnu grupu translacija, koja je konvolucioni modul&nbsp;nad &nbsp;Beurlingovom algebrom L¹_&omega;, gde je težina &omega; povezana sa operatorima translacije&nbsp;<br />prostora <em>E</em>. Banahov prostor <em>E^&#39;</em>_{&lowast;&nbsp;}označava prostor <em>L</em>¹_&omega;˅ &lowast; <em>E^&#39;</em>. Koristeći dobijene&nbsp;<br />rezultata proučavamo konvoluciju ultradistribucija. Prostori konvolutora &nbsp;<em>O^&#39;</em>^&lowast;_{<em>C&nbsp;</em>}(<strong>R</strong>ⁿ)&nbsp;temperiranih ultradistribucija, analizirani su pomoću dualnosti&nbsp;<br />test funkcijskih prostora <em>O</em>^&lowast;_<em>C</em> (<strong>R</strong>ⁿ), definisanih u ovoj tezi. Koristeći svojstva&nbsp;<br />translaciono - invarijantnih Banahovih prostora temperiranih ultradistribucija,&nbsp;<br />opet označenih sa <em>E</em>, dobijamo karakterizaciju konvolucije Romuieu-ovih &nbsp;ultradistribucija,&nbsp;preko integrabilnih ultradistribucija. Dokazujemo da: konvolucija&nbsp;<br />dve Roumieu-ove ultradistribucija <em>T</em>, <em>S</em> &isin; <em>D^&#39;</em>^{Mp}&nbsp;(<strong>R</strong>ⁿ) postoji ako i samo ako (&phi; &lowast; <em>S</em>ˇ)<em>T</em> &isin; <em>D^&#39;</em>^{Mp}&nbsp;_L¹ (<strong>R</strong>ⁿ) za svaki &phi; &isin; <em>D</em>^{Mp}(<strong>R</strong>ⁿ). Takođe, proučavamo granične vrednosti holomorfnih funkcija definisanih na tubama. Dokazane su nove teoreme &rdquo;otrog klina&rdquo;. Rezultati se zatim koriste za prezentaciju <em>D^&#39;_E^&#39;</em>_{&lowast;&nbsp;}preko faktor prostora holomorfnih funkcija. Takođe, data je prezentacija elemente <em>D</em>^&#39;_{<em>E^&#39;</em>&lowast;&nbsp;}koristeći heat kernel metode.</p>

<p>In the thesis are introduced and investigated spaces of Burling and of Roumieu type tempered ultradistributions, which are natural generalization of the space of Schwartz&rsquo;s tempered distributions in Denjoy-Carleman-Komatsu&rsquo;s theory of ultradistributions.&nbsp; It has been proved that the introduced spaces preserve all of the good properties Schwartz space has, among others, a remarkable one, that the Fourier transform maps continuposly the spaces into themselves.<br />In the first chapter the necessary notation and notions are given.<br />In the second chapter, the spaces of ultrarapidly decreasing ultradifferentiable functions and their duals, the spaces of Beurling and of Roumieu tempered ultradistributions, are introduced; their topological properties and relations with the known distribution and ultradistribution spaces and structural properties are investigated;&nbsp; characterization of&nbsp; the Hermite expansions&nbsp; and boundary value representation of the elements of the spaces are given.<br />The spaces of multipliers of the spaces of Beurling and of Roumieu type tempered ultradistributions are determined explicitly in the third chapter.<br />The fourth chapter is devoted to the investigation of&nbsp; Fourier, Wigner, Bargmann and Hilbert transforms on the spaces of Beurling and of Roumieu type tempered ultradistributions and their test spaces.<br />In the fifth chapter the equivalence of classical definitions of the convolution of Beurling type ultradistributions is proved, and the equivalence of, newly introduced definitions, of ultratempered convolutions of Beurling type ultradistributions is proved.<br />In the last chapter is given a necessary and sufficient condition for a convolutor of a space of tempered ultradistributions to be hypoelliptic in a space of integrable ultradistribution, is given, and hypoelliptic convolution equations are studied in the spaces.<br />Bibliograpy has 70 items.</p> / <p>U ovoj tezi su proučavani prostori temperiranih ultradistribucija Beurlingovog&nbsp; i Roumieovog tipa, koji su prirodna uop&scaron;tenja prostora Schwarzovih temperiranih distribucija u Denjoy-Carleman-Komatsuovoj teoriji ultradistribucija. Dokazano je ovi prostori imaju sva dobra svojstva, koja ima i Schwarzov prostor, izmedju ostalog, značajno svojstvo da Furijeova transformacija preslikava te prostore neprekidno na same sebe.<br />U prvom poglavlju su uvedene neophodne oznake i pojmovi.<br />U drugom poglavlju su uvedeni prostori ultrabrzo opadajucih ultradiferencijabilnih funkcija i njihovi duali, prostori Beurlingovih i Rumieuovih temperiranih ultradistribucija; proučavana su njihova topolo&scaron;ka svojstva i veze sa poznatim prostorima distribucija i ultradistribucija, kao i strukturne osobine; date su i karakterizacije Ermitskih ekspanzija i graničnih reprezentacija elemenata tih prostora.<br />Prostori multiplikatora Beurlingovih i Roumieuovih temperiranih ultradistribucija su okarakterisani u trećem poglavlju.<br />Četvrto poglavlje je posvećeno proučavanju Fourierove, Wignerove, Bargmanove i Hilbertove transformacije na prostorima Beurlingovih i Rouimieovih temperiranih ultradistribucija i njihovim test prostorima.<br />U petoj glavi je dokazana ekvivalentnost klasičnih definicija konvolucije na Beurlingovim prostorima ultradistribucija, kao i ekvivalentnost novouvedenih definicija ultratemperirane konvolucije ultradistribucija Beurlingovog tipa.<br />U poslednjoj glavi je dat potreban i dovoljan uslov da konvolutor prostora temperiranih ultradistribucija bude hipoeliptičan u prostoru integrabilnih ultradistribucija i razmatrane su neke konvolucione jednačine u tom prostoru.<br />Bibliografija ima 70 bibliografskih jedinica.</p>

Dans le cadre d'intersections de classes non quasi-analytiques à croissance modérée, J. Chaumat et A. M. Chollet ont démontré, notamment, un théorème d'extension de Whitney, pour des jets définis sur un compact et un théorème de Lojasiewicz sur la régulière situation. Ces intersections sont contenues dans l'intersection des classes de Gevrey. On établit ici un théorème d'extension dans une famille d'intersections de classes plus vaste, en ce sens que, tout jet de Whitney appartient à l'une des intersections considérées. Ensuite, en utilisant une méthode d'interpolation à l'aide de polynômes de Lagrange, due à W. Pawlucki et W. Plesniak, on établit aussi un théorème d'extension linéaire pour les jets définis sur des compacts ayant la propriété de Markov. Ces extensions de jets peuvent être choisies réelles analytiques sur le complémentaire du compact. Ces résultats sont complétés par trois exemples de situations pour lesquelles il n'existe pas d'opérateur d'extension linéaire continu. Enfin, on démontre un théorème de Lojasiewicz. Tous ces résultats sont étroitement reliés aux théorèmes classiques de la théorie des fonctions infiniment dérivables.

[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

<p>We study the expansions of the elements in <em>S</em>(ℝ₊^d) and <em>S</em>&#39;(ℝ₊^d) with respect to the Laguerre orthonormal basis. As a consequence, we obtain the Schwartz kernel theorem for <em>S</em>(ℝ₊^d) and <em>S</em>&#39;(ℝ₊^d). Also we give the extension theorem of Whitney type for <em>S</em>(ℝ₊^d). Next, we consider the G-type spaces i.e. the spaces <em>G</em>_{<em>&alpha;</em>}^{<em>&alpha;</em>}(ℝ₊^d), &alpha;&ge;1&nbsp; and their dual spaces which can be described as analogous to the Gelfand-Shilov spaces and their dual spaces. Actually, we show the exist-ence of the topological isomorphism between the <em>G</em>-type spaces and the subspaces of the Gelfand-Shilov spaces <em>S</em>_&alpha;/2^&alpha;/2(ℝ^d), &alpha;&ge;1&nbsp;consisting of &quot;even&quot; functions. Next, we show that the Fourier Laguerre coecients of the elements in the <em>G</em>-type spaces and their dual spaces characterize these spaces through the exponential and sub-exponentia l growth of the coecients. We provide the full topological description and the kernel theorem is proved. Also two structural theorems for the dual spaces of <em>G</em>-type spaces are obtained. Furthemore, we dene the new class of the Weyl pseudo-dierential operators with radial symbols belonging to the G-type spaces and their dual spaces. The continuity properties of this class of pseudo-dierential operators over the Gelfand-Shilov type spaces and their duals are proved. In this way the class of the Weyl pseudo-dierential operators is extended to the one with the radial symbols with the exponential and sub-exponential growth rate.</p> / <p>Proučavamo razvoje elemenata iz <em>S</em>(ℝ₊^d) i <em>S</em>&#39;(ℝ₊^d) preko Lagerove ortonormirane baze. Kao posledicu dobijamo &Scaron;varcovu teoremu o jezgru za preko Lagerove ortonormirane baze. Kao posledicu dobijamo &Scaron;varcovu teoremu o jezgru za <em>S</em>(ℝ₊^d) i <em>S</em>&#39;(ℝ₊^d). Takođe, pokazujemo i Teoremu Vitnijevog tipa za <em>S</em>(ℝ₊^d) . Zatim, posmatramo prostore G-tipa i.e. prostore <em>G</em>_&alpha;^&alpha;(ℝ^d), &alpha; &ge; 1 i njihove duale koji su analogni sa Geljfand-&Scaron;ilovim prostorima i njihovim dualima. Zapravo, pokazujemo da postoji topolo&scaron;ki izomorfizam između prostora <em>G</em>-tipa i potprostora Geljfand-&Scaron;ilovih prostora <em>S</em>_&alpha;/2^&alpha;/2(ℝ^d), &alpha; &ge; 1 koji sadrže &quot;parne&quot; funkcije. Dalje, dokazujemo da Furije Lagerovi koeficijenti elemenata iz prostora <em>G</em>-tipa i njihovih duala karakteri&scaron;u ove prostore kroz eksponencijalni i sub-eksponencijalni rast tih koeficijenata. Opisujemo topolo&scaron;ku strukturu ovih prostora i dajemo &Scaron;varcovu teoremu o jezgru. Takođe, dve strukturalne teoreme za duale prostora <em>G</em>-tipa su dobijene. Dalje, defini&scaron;emo novu klasu Vejlovih pseudo-diferencijalnih operatora sa radijalnim simbolima koji se nalaze u prostorima <em>G</em>-tipa i njihovim dualima. Pokazana je neprekidnost ove klase Vejlovih pseudo-diferencijalnih operatora na prostorima Geljfand-&Scaron;ilova i na njihovim dualima. Na ovaj način klasa Vejlovih pseudo-diferencijalnih operatora je pro&scaron;irena na radijalne simbole koji imaju eksponencijalni i sub-eksponencijalni rast.</p>