Дисертації з теми "Fourier and Schur multipliers"
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Zeng, Kai. "Some problems in harmonic analysis on twsited crossed products." Electronic Thesis or Diss., Bourgogne Franche-Comté, 2023. http://www.theses.fr/2023UBFCD048.
This thesis is devoted to the study of some problems in the harmonic analysis on twisted crossed products defined by twisted actions of a locally compact group G on a von Neumann algebra M. It consists of two parts. The first concerns twisted crossed products and their Fourier and Schur multipliers. We prove that the property of being QWEP for the twisted von Neumann algebra of a group G is independent of the underlying 2-cocycle and that the completely bounded Lp-Fourier multipliers on this twisted algebra are also independent of the 2-cocycle. Under the hypothesis of an amenable action, we establish several transference results between the Fourier and Schur multipliers on the noncommutative Lp spaces of the twisted crossed product.In the second part, we study Fourier multiplier commutators on the twisted crossed product of an Euclidean space. We characterize their Schatten p-class membership by that of their symbols in the associated Besov space. In addition, this part contains a formula on the Dixmier trace, which also gives us a characterization of the weak Schatten p-class membership of these commutators by a Sobolev space. In particular, our results apply to the case of quantum Euclidean spaces
McKee, Andrew. "Multipliers of dynamical systems." Thesis, Queen's University Belfast, 2017. https://pure.qub.ac.uk/portal/en/theses/multipliers-of-dynamical-systems(65b93a06-6e7b-420b-ae75-c28d373f8bdf).html.
Steen, Naomi Mary. "Unbounded generalisations of Schur and operator multipliers." Thesis, Queen's University Belfast, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.603070.
Coine, Clément. "Continuous linear and bilinear Schur multipliers and applications to perturbation theory." Thesis, Bourgogne Franche-Comté, 2017. http://www.theses.fr/2017UBFCD074/document.
In the first chapter, we define some tensor products and we identify their dual space. Then, we give some properties of Schatten classes. The end of the chapter is dedicated to the study of Bochner spaces valued in the space of operators that can be factorized by a Hilbert space.The second chapter is dedicated to linear Schur multipliers. We characterize bounded multipliers on B(Lp, Lq) when p is less than q and then apply this result to obtain new inclusion relationships among spaces of multipliers.In the third chapter, we characterize, by means of linear Schur multipliers, continuous bilinear Schur multipliers valued in the space of trace class operators. In the fourth chapter, we give several results concerning multiple operator integrals. In particular, we characterize triple operator integrals mapping valued in trace class operators and then we give a necessary and sufficient condition for a triple operator integral to define a completely bounded map on the Haagerup tensor product of compact operators. Finally, the fifth chapter is dedicated to the resolution of Peller's problems. We first study the connection between multiple operator integrals and perturbation theory for functional calculus of selfadjoint operators and we finish with the construction of counter-examples for those problems
Marcoci, Liviu-Gabriel. "A study of Schur multipliers and some Banach spaces of infinite matrices : /." Luleå : Department of Mathematics, Luleå University of Technology, 2010. http://pure.ltu.se/ws/fbspretrieve/4554227.
Akylzhanov, Rauan. "Lp-Lq Fourier multipliers on locally compact groups." Thesis, Imperial College London, 2018. http://hdl.handle.net/10044/1/60829.
Mathias, Maximilian [Verfasser], E. [Gutachter] Schmidt, and J. [Gutachter] Schur. "Über positive Fourier-Integrale / Maximilian Mathias ; Gutachter: E. Schmidt, J. Schur." Berlin : Humboldt-Universität zu Berlin, 2006. http://d-nb.info/1206192313/34.
Johnstone, Stephen. "Theory and applications of Fourier multipliers on locally compact groups." Thesis, University of Strathclyde, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.443141.
Marcoci, Liviu-Gabriel. "Some new results concerning Schur multipliers and duality results between Bergman-Schatten and little Bloch spaces /." Luleå : Luleå University of Technology, 2009. http://pure.ltu.se/ws/fbspretrieve/2732750.
Marcoci, Anca-Nicoleta. "Some new results concerning Lorentz sequence spaces and Schur multipliers : characterization of some new Banach spaces of infinite matrices." Licentiate thesis, Luleå : Luleå University of Technology, 2009. http://pure.ltu.se/ws/fbspretrieve/2727437.
Rodríguez, López Salvador. "Transference theory between quasi-Banach function spaces with applications to the restriction of Fourier multipliers." Doctoral thesis, Universitat de Barcelona, 2008. http://hdl.handle.net/10803/2118.
Kf= çk(x-y) f(y) dy
with k an L^1 function, the transferred operator T is defined by letting
Tf= çk(x-y) R_xf(y) dy.
Transfer methods deal with the study of the preservation of properties of K that are still valid for T, mostly focusing on the preservation of boundedness on Lebesgue spaces Lp. These methods has been applied to several problems in Mathematical Analysis, and especially to the problem of restrict Fourier multipliers to closed subgroups. These techniques have been extended by other authors as N. Asmar, E. Berkson and A. Gillespie, among many others. It is worth noting however, that these prior developments have always been focused on inequalities for operators on Lebesgue spaces Lp.
In this thesis there are developed several transference techniques for quasi-Banach spaces more general than Lebesgue spaces Lp, as Lorentz spaces Lp, q, Orlicz-Lorentz, Lorentz-Zygmund spaces as well as for weighted Lebesgue spaces Lp(w). The most significant applications are obtained in the field of restriction of Fourier multipliers for rearrangement invariant spaces and weighted Lebesgue spaces Lp(w). Specifically, we get generalizations of the results obtained by K. De Leeuw for Fourier multipliers. There are also developed similar techniques in the context of multilinear operators of convolution type, where the basic example is the bilinear Hilbert transform, as well as for modular inequalities and inequalities arising in extrapolation
Wang, Simeng. "Some problems in harmonic analysis on quantum groups." Thesis, Besançon, 2016. http://www.theses.fr/2016BESA2062/document.
This thesis studies some problems in the theory of harmonic analysis on compact quantum groups. It consists of three parts. The first part presents some elementary Lp theory of Fourier transforms, convolutions and multipliers on compact quantum groups, including the Hausdorff-Young theory and Young’s inequalities. In the second part, we characterize positive convolution operators on a finite quantum group G which are Lp-improving, and also give some constructions on infinite compact quantum groups. The methods for ondegeneratestates yield a general formula for computing idempotent states associated to Hopf images, which generalizes earlier work of Banica, Franz and Skalski. The third part is devoted to the study of Sidon sets, _(p)-sets and some related notions for compact quantum groups. We establish several different characterizations of Sidon sets, and in particular prove that any Sidon set in a discrete group is a strong Sidon set in the sense of Picardello. We give several relations between Sidon sets, _(p)-sets and lacunarities for Lp-Fourier multipliers, generalizing a previous work by Blendek and Michali˘cek. We also prove the existence of _(p)-sets for orthogonal systems in noncommutative Lp-spaces, and deduce the corresponding properties for compact quantum groups. Central Sidon sets are also discussed, and it turns out that the compact quantum groups with the same fusion rules and the same dimension functions have identical central Sidon sets. Several examples are also included. The thesis is principally based on two works by the author, entitled “Lp-improvingconvolution operators on finite quantum groups” and “Lacunary Fourier series for compact quantum groups”, which have been accepted for publication in Indiana University Mathematics Journal and Communications in Mathematical Physics respectively
Neuwirth, Stefan. "Multiplicateurs et analyse fonctionnelle." Phd thesis, Université Pierre et Marie Curie - Paris VI, 1999. http://tel.archives-ouvertes.fr/tel-00010399.
Arhancet, Cédric. "Estimation de normes dans les espaces Lp non commutatifs et applications." Phd thesis, Université de Franche-Comté, 2011. http://tel.archives-ouvertes.fr/tel-00647348.
Yin, Zhi. "Espaces de Hardy en probabilités et analyse harmonique quantiques." Phd thesis, Université de Franche-Comté, 2012. http://tel.archives-ouvertes.fr/tel-00838496.
Miraglio, Pietro. "Estimates and rigidity for stable solutions to some nonlinear elliptic problems." Doctoral thesis, Universitat Politècnica de Catalunya, 2020. http://hdl.handle.net/10803/668832.
Mi tesis se encaja en el estudio de las EDPs elípticas. Está dividida en dos partes: la primera trata una ecuación no-lineal con el p-Laplaciano, la segunda de un problema no-local. En la primera parte, estudiamos la regularidad de las soluciones estables de una ecuación no lineal con el p-Laplaciano en un dominio acotado. Esta ecuacion es la versión no-lineal de la ámpliamente estudiada ecuacion semilineal con el Laplaciano. Cabré, Figalli, Ros-Oton, y Serra han demostrado recientemente que las soluciones estables de las ecuaciones semilineales son acotadas, y por tanto regulares, hasta la dimensión 9. Este resultado es optimal. En el caso del p-Laplaciano, la regularidad de las soluciones estables se conjetura de ser cierta hasta una dimension critica y, de hecho, se conocen ejemplos de soluciones no acotadas cuando la dimension llega al valor critico. Además, se ha demostrado que en el caso radial o assumiendo hipótesis fuertes sobre la no-linealidad las soluciones estables son acotadas hasta la dimension critica. En el primer capítulo, demostramos que las soluciones estables son acotadas, bajo una nueva condición en n y p, que es optimal en el caso radial, y más restrictiva en el caso general. Esta investigación mejora conocidos resultados del tema y es el primer ejemplo, para el p-Laplaciano, de un método que produce un resultado para el caso general y un resultado optimal en el caso radial. En la primera parte, nos ocupamos también de las desigualdades funcionales del tipo Hardy y Sobolev sobre hipersuperfícies del espacio Euclideo, todas conteniendo un término de curvatura media. Nuestra motivación proviene de varias apliaciones que tienen estas desigualdades en el estudio de estimaciones para las soluciones estables. En detalle, damos una demostración simple de la conocida desigualdad de Michael-Simon y Allard, obtenemos dos formas nuevas de la desigualdad de Hardy sobre hipersuperfícies, y otra desigualdad de Hardy-Poincaré. En la segunda parte, nos ocupamos de un problema de Dirichlet-Neumann que emerge de un modelo para las ondas en el agua. El sistema se describe con una ecuación de difusión en una tira de altura fija, que contiene un parámetro a en (-1,1). La parte superior de la tira es dotada de una condicion 0 de Neumann, mientras en la parte inferior tenemos un dato de Dirichlet y una ecuación con una nonlinearidad regular. Este problema puede ser reformulado como una ecuación no-local sobre la componente dotada del dato de Dirichlet, definiendo un operador de Dirichlet-Neumann apropiado. Primero, demostramos un teorema del tipo Liouville, que garantiza la simetría unidimensional de las soluciones monótonas, asumiendo un control sobre el crecimiento de la energía asociada. Como consecuencia, obtenemos la simetría 1D de las soluciones estables en dimension 2. Para n=3, obtenemos estimaciónes optimales de la energía para las soluciones que minimizan la energía y para las soluciones monótonas. Estas estimaciones nos conducen a la simetría 1D de estas clases de soluciones, aplicando nuestro teorema del tipo Liouville. Relativo a este problema, estudiamos también la naturaleza del operador de Dirichlet-Neumann. Primero, deducimos su expresión como operador de Fourier, que anteriormente solo se conocía para a=0. Este resultado evidencia la naturaleza del operador, que es no-local pero no puramente fraccionaria. Estudiamos en profundidad este comportamiento mixto del operador a través del estudio de la G-convergencia de un funcional energía asociado al operador. Demostramos la G-convergencia de nuestro funcional a un límite que corresponde a una energía de interacción pura cuando a en (0,1) y al perímetro clásico cuando a en (-1,0]. El límite a=0, así como el G-límite para el régimen a en (-1,0], es común a otros problemas no-locales tratados en la literatura. Al contrario, el funcional límite en el régimen puramente no-local es nuevo y diferente a otros funciona
Questa tesi si occupa di equazioni differenziali alle derivate parziali di tipo ellittico. È divisa in due parti: la prima riguarda un’equazione nonlineare per il p-Laplaciano, mentre la seconda è incentrata su un problema nonlocale, che può essere formulato per mezzo di un operatore di Dirichlet-Neumann collegato con il Laplaciano frazionario. Nella prima parte, studiamo la regolarità delle soluzioni stabili dell’equazione nonlineare per il p-Laplaciano dove W è un dominio limitato, p 2 (1,+¥) e f è una nonlinearità C1. Questa equazione è la versione nonlineare dell’equazione semilineare ������������Du = f (u) in un dominio limitato W Rn, che è stata ampiamente studiata in letteratura. Molto recentemente, Cabré, Figalli, Ros-Oton, e Serra [38] hanno dimostrato che le soluzioni stabili delle equazioni semilineari sono limitate, e quindi regolari, in dimensione n 9. Questo risultato è ottimale, dato che esempi di soluzioni illimitate e stabili sono noti in dimensione n 10. Inoltre, i risultati in [38] forniscono una risposta completa ad un annoso problema aperto, proposto da Brezis e Vázquez [25], sulla regolarità delle soluzioni estremali dell’equazione ������������Du = l f (u). Queste ultime sono infatti esempi non banali di soluzioni stabili di equazioni semilineari, che possono essere limitate o illimitate in dipendenza della dimensione n, del dominio W, e della nonlinearità f . In questa tesi studiamo la limitatezza delle soluzioni stabili di (0.4), che si congettura essere vera fino alla dimensione n < p + 4p/(p ������������ 1). Sono infatti noti esempi di soluzioni stabili e illimitate quando n p + 4p/(p ������������ 1), anche quando il dominio è la palla unitaria. Inoltre, nel caso radiale o assumendo ipotesi forti sulla nonlinearità, è stato dimostrato che le soluzioni stabili di (0.4) sono limitate quando n < p + 4p/(p ������������ 1). Nel Capitolo 1 della tesi dimostriamo una nuova stima L¥ a priori per le soluzioni stabili di (0.4), assumendo una nuova condizione su n e p, che è ottimale nel caso radiale e più restrittiva nel caso generale. Il nostro risultato migliora ciò che è noto in letteratura e ed è il primo esempio di tecnica che produce sia un risultato nel caso non radiale sia il risultato ottimale nel caso radiale. Per ottenere questo risultato estendiamo al caso del p-Laplaciano una tecnica sviluppata da Cabré [30] per il caso classico del problema, con p = 2. La strategia si basa su una disuguaglianza di Hardy sugli insiemi di livello della soluzione, combinata con una disuguaglianza di tipo geometrico per le soluzioni stabili di (0.4). Nella prima parte della tesi ci occupiamo anche di disuguaglianze funzionali di tipo Hardy e Sobolev, su ipersuperfici dello spazio euclideo. Nel fare ciò siamo motivati dalle varie applicazioni di questo tipo di risultati allo studio di stime a priori per le soluzioni stabili, sia nel caso semilineare che nel caso nonlineare ...
Wang, Xumin. "Functional and harmonic analysis of noncommutative Lp spaces associated to compact quantum groups." Thesis, Bourgogne Franche-Comté, 2019. http://www.theses.fr/2019UBFCD040.
This thesis is devoted to studying the analysis on compact quantum groups. It consists of two parts. First part presents the classification of invariant quantum Markov semigroups on these quantum homogeneous spaces. The generators of these semigroups are viewed as Laplace operators on these spaces.The classical sphere, the free sphere, and the half-liberated sphere are considered as examples and the generators of Markov semigroups on these spheres are classified. We compute spectral dimensions for the three families of spheres based on the asymptotic behavior of the eigenvalues of their Laplace operator.In the second part, we study of convergence of Fourier series for non-abelian groups and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as some summation methods and mean convergence of associated noncommutative Fourier series. We establish a general criterion of maximal inequalities for approximative identities of noncommutative Fourier multipliers. As a result, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions, so that the associated Fourier multipliers on noncommutative Lp-spaces satisfy the pointwise convergence. Our results also apply to the almost everywhere convergence of Fourier series of Lp-functions on non-abelian compact groups. On the other hand, we obtain the dimension free bounds of noncommutative Hardy-Littlewood maximal inequalities in the operator-valued Lp space associated with convex bodies
Zhang, Haonan. "Some problems in noncommutative analysis." Thesis, Bourgogne Franche-Comté, 2019. http://www.theses.fr/2019UBFCD043.
This PhD thesis is devoted to the study of some problems in noncommutative analysis. It consists of four parts, ranging from quantum groups and noncommutative harmonic analysis to quantum information. Firstly, we decide all the idempotent states on Sekine quantum groups, which is achieved by solving a system of equations using linear algebras and elementary number theory. This answers a question of Franz and Skalski stated in 2009. Secondly, we study the infinitely divisible states on finite quantum groups, i.e., states that admit n-th root for all nge 1. We show that every infinitely divisible state on a finite quantum group is of Poisson type, that is, it can be represented as an exponential relative to some idempotent state. Thirdly, we give two sufficient conditions for boundedness of L_p-Fourier multipliers on discrete group von Neumann algebras. Very few of such results were known before. Our idea is the observation that in the discrete case it suffices to consider L_p-L_q Fourier multipliers. Finally, in the area of quantum information, we confirm a conjecture of Carlen, Frank and Lieb (and then a weaker conjecture of Audenaert and Datta). As a consequence, we identify all the pairs (alpha,z) such that the alpha-z Rényi relative entropy is monotone under completely positive trace preserving maps, or satisfies Data Processing Inequality. The key part of the proof is a modification of a widely-used variational method. Its power yields simple proofs of many known results
MIRAGLIO, PIETRO. "ESTIMATES AND RIGIDITY FOR STABLE SOLUTIONS TO SOME NONLINEAR ELLIPTIC PROBLEMS." Doctoral thesis, Università degli Studi di Milano, 2020. http://hdl.handle.net/2434/704717.
This thesis deals with the study of elliptic PDEs. The first part of the thesis is focused on the regularity of stable solutions to a nonlinear equation involving the p-Laplacian, in a bounded domain of the Euclidean space. The technique is based on Hardy-Sobolev inequalities in hypersurfaces involving the mean curvature, which are also investigated in the thesis. The second part concerns, instead, a nonlocal problem of Dirichlet-to-Neumann type. We study the one-dimensional symmetry of some subclasses of stable solutions, obtaining new results in dimensions n=2, 3. In addition, we carry out the study of the asymptotic behaviour of the operator associated with this nonlocal problem, using Γ-convergence techniques.
Dušanka, Perišić. "On Integral Transforms and Convolution Equations on the Spaces of Tempered Ultradistributions." Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 1992. https://www.cris.uns.ac.rs/record.jsf?recordId=73337&source=NDLTD&language=en.
U ovoj tezi su proučavani prostori temperiranih ultradistribucija Beurlingovog i Roumieovog tipa, koji su prirodna uopštenja 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.U prvom poglavlju su uvedene neophodne oznake i pojmovi.U drugom poglavlju su uvedeni prostori ultrabrzo opadajucih ultradiferencijabilnih funkcija i njihovi duali, prostori Beurlingovih i Rumieuovih temperiranih ultradistribucija; proučavana su njihova topološka 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.Prostori multiplikatora Beurlingovih i Roumieuovih temperiranih ultradistribucija su okarakterisani u trećem poglavlju.Četvrto poglavlje je posvećeno proučavanju Fourierove, Wignerove, Bargmanove i Hilbertove transformacije na prostorima Beurlingovih i Rouimieovih temperiranih ultradistribucija i njihovim test prostorima.U petoj glavi je dokazana ekvivalentnost klasičnih definicija konvolucije na Beurlingovim prostorima ultradistribucija, kao i ekvivalentnost novouvedenih definicija ultratemperirane konvolucije ultradistribucija Beurlingovog tipa.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.Bibliografija ima 70 bibliografskih jedinica.
Bagchi, Sayan. "Weighted Norm Inequalities for Weyl Multipliers and Hermite Pseudo-Multipliers." Thesis, 2015. http://etd.iisc.ac.in/handle/2005/3641.
Bagchi, Sayan. "Weighted Norm Inequalities for Weyl Multipliers and Hermite Pseudo-Multipliers." Thesis, 2015. http://etd.iisc.ernet.in/2005/3641.
Popa, Ana-Maria. "On completely bounded multipliers of the Fourier algebra A(G) /." 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3337888.
Source: Dissertation Abstracts International, Volume: 69-11, Section: B, page: 6848. Adviser: Zhong-Jin Ruan. Includes bibliographical references (leaves 77-80) Available on microfilm from Pro Quest Information and Learning.
Wu, Chong-Chou, and 吳忠洲. "Pipeline Fast Fourier Transform Processors Realization with Various Complex Multipliers." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/60415564356231341991.
國立高雄應用科技大學
電子工程系
98
Due to the popularity of the communication systems, the Fourier transform is still one of research and development topics of wired and wireless communication. The high-speed computing of the discrete Fourier transform is very important in the real-time signal processing system. So many fast Fourier transform (FFT) algorithm are developed. Because of the regularity of FFT algorithm, it is very suitable for the implementation by using hardware circuits. Most of the developed algorithms reduce the computational complexity. In this thesis, we use radix-22 algorithm which can reduce the computational complexity from to . In this study, we compare various circuit architectures of fast Fourier transform in the view of hardware regularity, needed memory space and the number of computing operation. Finally, we adopt the radix-22 algorithm and the single-path delay feedback (SDF) architecture to implement the high-performance FFT processor. The conventional complex multiplier, multiplier-less canonical signed digit (CSD) complex multiplier and coordinate rotation digital computer (Cordic) architecture are used to realize pipelined fast Fourier transform processors. To reduce the error, we also realize our complex multiplier computing circuits with the double rounding technique. The Xilinx ISE software is used to synthesis the hardware, it shows that the 16-point FFT processor based on the multiplier-less canonical signed digit (CSD) complex multiplier can achieve the advantages of less needed hardware and moderate accuracy.
Kazaniecki, Krystian. "Analytic properties of operators on the non-reflexive spaces of smooth functions." Doctoral thesis, 2019. https://depotuw.ceon.pl/handle/item/3344.
Moja rozprawa doktorska składa się z rezultatów, które uzyskałem badając własności operatorów na niere eksywnych przestrzeniach funkcji gładkich. W dziedzinie analizy funkcjonalnej jednymi z najbardziej interesujących przykładów przestrzeni są przestrzenie funkcji analitycznych (n.p. przestrzenie Hardy’ego) i przestrzenie funkcji gładkich (n.p. przestrzenie Sobolewa, przestrzenie Biesowa). W odróżnieniu od przestrzeni Hardy’ego, gdzie własności operatorów są dobrze zbadane, nasza wiedza na temat własności operatorów w nierefleksywnych przestrzeniach Sobolewa jest wciąż niezadowalająca.
Medalha, Samuel João Baltazar. "Algebras of Convolution Type Operators on Weighted Variable Lebesgue Spaces." Master's thesis, 2021. http://hdl.handle.net/10362/135865.
Provamos uma versão do teorema de interpolação de Riesz-Thorin para alguns tipos de espaços de Lebesgue com expoente variável e peso. De forma a atingir este objectivo, usamos a teoria desenvolvida por Calderón no seu artigo de 1964. Usando a versão do teorema de Riesz-Thorin obtida, provamos uma versão da desigualdade de Stechkin para espaços de Lebesgue com expoente variável e peso. Isto permite-nos definir álgebras de multiplicadores de Fourier associados a funções de variação limitada. Após analisada a invertibilidade dos operadores de convolução com símbolos contínuos por troços, deslocamos a nossa atenção para multiplicadores de Fourier fracamente oscilantes. Terminamos com a prova de que a imagem na álgebra de Calkin da álgebra de operadores tipo convolução com dados fracamente oscilantes é comutativa.