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Dissertations / Theses on the topic 'Integral operators'

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Published: 4 June 2021
Last updated: 30 July 2024

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1

Chunaev, Petr. "Singular integral operators and rectifiability." Doctoral thesis, Universitat Autònoma de Barcelona, 2018. http://hdl.handle.net/10803/663827.

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Los problemas que estudiamos en esta tesis se encuentran en el área de Análisis Armónico y Teoría de la Medida Geométrica. En particular, consideramos la conexión entre las propiedades analíticas de operadores integrales singulares definidos en $L^2(\mu)$ y asociados con algunos núcleos de Calderón-Zygmund y las propiedades geométricas de la medida $\mu$. Seamos más precisos. Sea $E$ un conjunto de Borel en el plano complejo con la medida lineal de Hausdorff $H^1$ finita y distinta de cero, es decir, $00$ es una pequeña constante absoluta. Es importante que, para algunos de los $t$ que acabamos de mencionar, el llamado método de curvatura comúnmente utilizado para relacionar $L^2$-acotación y rectificabilidad no está disponible, pero todavía es posible establecer la propiedad mencionada. Hasta donde sabemos, es el primer ejemplo de este tipo en el plano complejo. También vale la pena mencionar que ampliamos nuestros resultados a una clase aún más general de núcleos y, además, consideramos problemas análogos para conjuntos $E$ Ahlfors-David-regulares.
The problems that we study in this thesis lie in the area of Harmonic Analysis and Geometric Measure Theory. Namely, we consider the connection between the analytic properties of singular integral operators defined in $L^2(\mu)$ and associated with some Calderón-Zygmund kernels and the geometric properties of the measure $\mu$. Let us be more precise. Let $E$ be a Borel set in the complex plane with non-vanishing and finite linear Hausdorff measure $H^1$, i.e. such that $00$ is a small absolute constant. It is important that for some of the $t$ just mentioned the so called curvature method commonly used to relate $L^2$-boundedness and rectifiability is not available but it is still possible to establish the above-mentioned property. To the best of our knowledge, it is the first example of this type in the plane. It is also worth mentioning that we extend our results to even more general class of kernels and additionally consider analogous problems for Ahlfors-David regular sets $E$.
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2

Vaktnäs, Marcus. "On Singular Integral Operators." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-355872.

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3

Beil, Joel S. "Geometric Properties of Orbits of Integral Operators." Kent State University / OhioLINK, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=kent1270503593.

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4

Khan, Mumtaz Ahmad. "On fractional integral operators of three variables and integral transforms." Pontificia Universidad Católica del Perú, 2012. http://repositorio.pucp.edu.pe/index/handle/123456789/96049.

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The present paper is a continuation to authors paper [11]where three variable analogues of certain fractional integraloperators of M. Saigo were investigated. This paper dealswith the effect of operating three variable analogues of Mellinand Laplace transforms on these three variable analogues offractional integral operators of the earlier paper.
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5

González, Luiz Felipe. "Some integral operators in thermodynamic formalism." Thesis, University of Warwick, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.396693.

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6

Hartung, Tobias. "z-functions of Fourier Integral Operators." Thesis, King's College London (University of London), 2015. http://kclpure.kcl.ac.uk/portal/en/theses/zfunctions-of-fourier-integral-operators(933e5068-2890-4d32-a991-fafa784bfda7).html.

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Based on Guillemin’s work on gauged Lagrangian distributions, we will introduce the notion of a gauged poly-log-homogeneous distribution as an approach to ζ-functions for a class of Fourier Integral Operators which includes cases of amplitudes with asymptotic expansion Σk∈N amk where each amk is log-homogeneous with degree of homogeneity mk but violating R(mk) → −∞. We will calculate the Laurent expansion for the ζ-function and give formulae for the coefficients in terms of the phase function and amplitude, as well as investigate generalizations to the Kontsevich-Vishik trace. Using stationary phase approximation, series representations for the Laurent coefficients and values of ζ-functions will be stated explicitly, and the kernel singularity structure will be studied. This will yield algebras of Fourier Integral Operators which purely consist of Hilbert-Schmidt operators and whose ζ-functions are entire, as well as algebras in which the generalized Kontsevich- Vishik trace is form-equivalent to the pseudo-differential operator case. Additionally, we will introduce an approximation method (mollification) for ζ-functions of Fourier Integral Operators whose amplitudes are poly-log-homogeneous at zero by ζ-functions of Fourier Integral Operators with “regular” amplitudes. In part II, we will study Bochner-, Lebesgue-, and Pettis integration in algebras of Fourier Integral Operators. The integration theory will extend the notion of parameter dependent Fourier Integral Operators and is compatible with the Atiyah-Jänich index bundle as well as the ζ-function calculus developed in part I. Furthermore, it allows one to emulate calculations using holomorphic functional calculus in algebras without functional calculus, and to consider measurable families of Fourier Integral Operators as they appear, for instance, in heat- and wave-traces of manifolds whose metrics are subject to random (possibly singular) perturbations.
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7

Khan, Mumtaz Ahmad, and Abukhammash Ghazi Salama. "On certain fractional integral operators of two variables and integral transforms." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/97226.

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The present paper is in continuation to authors earlier paper {9} where two variable analogues of certain fractional integral operators of M. Saigo were investigated. This paper deals with the effect of operating two variable analogues of Mellin and Laplace transforms on these two variable analogues of fractional integral operators of the earlier paper.
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8

Reguera, Rodriguez Maria del Carmen. "Sharp weighted estimates for singular integral operators." Diss., Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/39522.

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The thesis provides answers, in one case partial and in the other final, to two conjectures in the area of weighted inequalities for Singular Integral Operators. We study the mapping properties of these operators in weighted Lebesgue spaces with weight w. The novelty of this thesis resides in proving sharp dependence of the operator norm on the Muckenhoupt constant associated to the weigth w for a rich class of Singular Integral operators. The thesis also addresses the end point case p=1, providing counterexamples for the dyadic and continuous settings.
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9

Huettenmueller, Rhonda. "The Pettis Integral and Operator Theory." Thesis, University of North Texas, 2001. https://digital.library.unt.edu/ark:/67531/metadc2844/.

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Let (Ω, Σ, µ) be a finite measure space and X, a Banach space with continuous dual X*. A scalarly measurable function f: Ω→X is Dunford integrable if for each x* X*, x*f L1(µ). Define the operator Tf. X* → L1(µ) by T(x*) = x*f. Then f is Pettis integrable if and only if this operator is weak*-to-weak continuous. This paper begins with an overview of this function. Work by Robert Huff and Gunnar Stefansson on the operator Tf motivates much of this paper. Conditions that make Tf weak*-to-weak continuous are generalized to weak*-to­weak continuous operators on dual spaces. For instance, if Tf is weakly compact and if there exists a separable subspace D X such that for each x* X*, x*f = x*fχDµ-a.e, then f is Pettis integrable. This nation is generalized to bounded operators T: X* → Y. To say that T is determined by D means that if x*| D = 0, then T (x*) = 0. Determining subspaces are used to help prove certain facts about operators on dual spaces. Attention is given to finding determining subspaces far a given T: X* → Y. The kernel of T and the adjoint T* of T are used to construct determining subspaces for T. For example, if T*(Y*) ∩ X is weak* dense in T*(Y*), then T is determined by T*(Y*) ∩ X. Also if ker(T) is weak* closed in X*, then the annihilator of ker(T) (in X) is the unique minimal determining subspace for T.
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10

Ehrhardt, Torsten. "Factorization theory for Toeplitz plus Hankel operators and singular integral operators with flip." Doctoral thesis, [S.l. : s.n.], 2004. http://deposit.ddb.de/cgi-bin/dokserv?idn=972573305.

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11

Santana, Edixon Manuel Rojas. "A study of singular integral operators with shift." Doctoral thesis, Universidade de Aveiro, 2010. http://hdl.handle.net/10773/3882.

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Doutoramento em Matemática
Nesta tese, consideram-se operadores integrais singulares com a acção extra de um operador de deslocacamento de Carleman e com coeficientes em diferentes classes de funções essencialmente limitadas. Nomeadamente, funções contínuas por troços, funções quase-periódicas e funções possuíndo factorização generalizada. Nos casos dos operadores integrais singulares com deslocamento dado pelo operador de reflexão ou pelo operador de salto no círculo unitário complexo, obtêm-se critérios para a propriedade de Fredholm. Para os coeficientes contínuos, uma fórmula do índice de Fredholm é apresentada. Estes resultados são consequência das relações de equivalência explícitas entre aqueles operadores e alguns operadores adicionais, tais como o operador integral singular, operadores de Toeplitz e operadores de Toeplitz mais Hankel. Além disso, as relações de equivalência permitem-nos obter um critério de invertibilidade e fórmulas para os inversos laterais dos operadores iniciais com coeficientes factorizáveis. Adicionalmente, aplicamos técnicas de análise numérica, tais como métodos de colocação de polinómios, para o estudo da dimensão do núcleo dos dois tipos de operadores integrais singulares com coeficientes contínuos por troços. Esta abordagem permite também a computação do inverso no sentido Moore-Penrose dos operadores principais. Para operadores integrais singulares com operadores de deslocamento do tipo Carleman preservando a orientação e com funções contínuas como coeficientes, são obtidos limites superiores da dimensão do núcleo. Tal é implementado utilizando algumas estimativas e com a ajuda de relações (explícitas) de equivalência entre operadores. Focamos ainda a nossa atenção na resolução e nas soluções de uma classe de equações integrais singulares com deslocamento que não pode ser reduzida a um problema de valor de fronteira binomial. De forma a atingir os objectivos propostos, foram utilizadas projecções complementares e identidades entre operadores. Desta forma, as equações em estudo são associadas a sistemas de equações integrais singulares. Estes sistemas são depois analisados utilizando um problema de valor de fronteira de Riemann. Este procedimento tem como consequência a construção das soluções das equações iniciais a partir das soluções de problemas de valor de fronteira de Riemann. Motivados por uma grande diversidade de aplicações, estendemos a definição de operador integral de Cauchy para espaços de Lebesgue sobre grupos topológicos. Assim, são investigadas as condições de invertibilidade dos operadores integrais neste contexto.
In this thesis we consider singular integral operators with the extra action of a Carleman shift operator and having coefficients on different classes of essentially bounded functions. Namely, continuous, piecewise continuous, semi-almost periodic and generalized factorable functions. In the cases of the singular integral with shift action given by the reflection or the flip operator on the complex unit circle, we obtain a Fredholm criteria and, for the continuous coefficients case, an index formula is also provided. These results are consequence of explicit equivalence operator relations between those operators and some extra operators such as pure singular integral, Toeplitz and Toeplitz plus Hankel operators. Furthermore, the equivalence relations allow us to give an invertibility criterion and formulas for the left-sided and right-sided inverses of the initial operators with generalized factorable coefficients. In addition, we apply numerical analysis techniques, as polynomial collocation methods, for the study of the kernel dimension of these two kinds of singular integral operators with piecewise continuous coefficients. This approach also permits us to compute the Moore-Penrose inverse of the main operators. For singular integral operators with generic preserving-orientation Carleman shift operators and continuous functions as coefficients, upper bounds for the kernel dimensions are obtained. This is implemented by using some estimations which are derived with the help of certain explicit operator relations. We also focus our attention to the solvability, and the solutions, of a class of singular integral equations with shift which cannot be reduced to a binomial boundary value problem. To attain our goals, some complementary projections and operator identities are used. In this way, the equations under study are associated with systems of pure singular integral equations. These systems will be then analyzed by means of a corresponding Riemann boundary value problem. As a consequence of such a procedure, the solutions of the initial equations are constructed from the solutions of Riemann boundary value problems. Motivated by a large diversity of applications, we extend the definition of Cauchy integral operator to the framework of Lebesgue spaces on topological groups. Thus, invertibility conditions for paired operators in this setting are investigated.
FCT - SFRH/BD/30679/2006
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12

Andrews, Graeme David. "SG fourier integral operators with closure under composition." Thesis, Imperial College London, 2009. http://hdl.handle.net/10044/1/11990.

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13

Vähäkangas, Antti V. "Boundedness of weakly singular integral operators on domains /." Helsinki : Suomalainen Tiedeakatemia, 2009. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=018603140&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.

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14

Rogers, Keith McKenzie School of Mathematics UNSW. "Real and p-adic oscillatory integrals." Awarded by:University of New South Wales. School of Mathematics, 2004. http://handle.unsw.edu.au/1959.4/20521.

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After our introduction in Chapter 1, we consider van der Corput's lemma in Chapter 2. We find the nodes that minimize divided differences, and use these to find the sharp constant in a related sublevel set estimate. We go on to find the sharp constant in the first instance of the van der Corput lemma using a complex mean value theorem for integrals. With these bounds we improve the constant in the general van der Corput lemma, so that it is asymptotically sharp. In Chapter 3 we review the p-adic numbers and some results from Fourier analysis over the p-adics. In Chapter 4 we prove a p-adic version of van der Corput's lemma for polynomials, opening the way for the study of oscillatory integrals over the p-adics. In Chapter 5 we apply this result to bound maximal averages. We show that maximal averages over curves defined by p-adic polynomials are Lq bounded, where 1<q<infinity
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15

Doll, Moritz [Verfasser]. "Fourier integral operators on non-compact manifolds / Moritz Doll." Hannover : Gottfried Wilhelm Leibniz Universität Hannover, 2018. http://d-nb.info/116631393X/34.

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16

Robinson, Helen Jane. "Theorems of Helson-Szegö type and iterated integral operators." Thesis, University of York, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.428423.

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17

Li, Xiaochun. "Uniform bounds for the bilinear Hilbert transforms /." free to MU campus, to others for purchase, 2001. http://wwwlib.umi.com/cr/mo/fullcit?p3025634.

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18

Battisti, Ubertino [Verfasser]. "Zeta functions of pseudodifferential operators and Fourier integral operators on manifolds with boundary / Ubertino Battisti." Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2012. http://d-nb.info/1024816648/34.

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19

Peat, Rhona Margaret. "Fractional powers of operators and mellin multipliers." Thesis, University of Strathclyde, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.366801.

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20

Khan, Mumtaz Ahmad, and Bhagwat Swaroop Sharma. "A study of three variable analogues of certain fractional integral operators." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/95821.

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The paper deals with a three variable analogues of certain fractional integral operators introduced by M. Saigo. Resides giving three variable analogues of earlier known fractional integral operators of one variable as a special cases of newly defined operators, the paper establishes certain results in the form of theorems including integration by parts.
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21

Coriasco, Sandro, and Panarese Paolo. "Fourier integral operators defined by classical symbols with exit behaviour." Universität Potsdam, 2000. http://opus.kobv.de/ubp/volltexte/2008/2589/.

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We continue the investigation of the calculus of Fourier Integral Operators (FIOs) in the class of symbols with exit behaviour (SG symbols). Here we analyse what happens when one restricts the choice of amplitude and phase functions to the subclass of the classical SG symbols. It turns out that the main composition theorem, obtained in the environment of general SG classes, has a "classical" counterpart. As an application, we study the Cauchy problem for classical hyperbolic operators of order (1, 1); for such operators we refine the known results about the analogous problem for general SG hyperbolic operators. The material contained here will be used in a forthcoming paper to obtain a Weyl formula for a class of operators defined on manifolds with cylindrical ends, improving the results obtained in [9].
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22

Arendarenko, Larissa. "Some new Hardy-type Inequalities for integral operators with kernels." Licentiate thesis, Luleå tekniska universitet, Matematiska vetenskaper, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-26661.

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This Licentiate thesis deals with the theory of Hardy-type inequalities in anew situation, namely when the classical Hardy operator is replaced by amore general operator with kernel. The kernels we consider belong to thenew classes O+ n and O-n , n = 0; 1; :::, which are wider than co-called Oinarovclass of kernels.The thesis consists of three papers (papers A, B and C), an appendix topaper A and an introduction, which gives an overview to this specific fieldof functional analysis and also serves to put the papers in this thesis into amore general frame.In paper A some new Hardy-type inequalities for the case with Hardy-Volterra integral operators involved are proved and discussed. The case 1

Godkänd; 2011; 20111114 (larare); LICENTIATSEMINARIUM Ämnesområde: Matematik/Mathematics Examinator: Professor Lars-Erik Persson, Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet Diskutant: Professor Massimo Lanza de Cristoforis, Dipartamento di Matematica, Universita degli Studi di Padova, Italy Tid: Tisdag den 20 december 2011 kl 10.00 Plats: D2214-15, Luleå tekniska universitet

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23

Alzahrani, Faris. "A study of differential and integral operators in linear viscoelasticity." Thesis, Cardiff University, 2013. http://orca.cf.ac.uk/53333/.

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This thesis identifies and explores a link between the theory of linear viscoelasticity and the spectral theory of Sturm-Liouville problems. The thesis is divided into five chapters. Chapter 1 gives a brief account of the relevant parts of the theory of linear viscoelasticity and lays the foundation for making the link with spectral theory. Chapter 2 is concerned with the construction of approximate Dirichlet series for completely monotonic functions. The chapter introduces various connections between non-negative measures, orthogonal polynomials, moment problems, and the Stieltjes continued fraction. Several interlacing properties for discrete relaxation and retardation times are also proved. The link between linear viscoelasticity and spectral theory is studied in detail in Chapter 3. The stepwise spectral functions associated with some elementary viscoelastic models are derived and their Sturm-Liouville potentials are explicitly found by using the Gelfand-Levitan method for inverse spectral problems. Chapter 4 presents a new family of exact solutions to the nonlinear integrodifferential A-equation, which is the main equation in a recent method proposed by Barry Simon for solving inverse spectral problems. Starting from the A-amplitude A(t) = A(t, 0) which is determined by the spectral function, the solution A(t, x) of the A-equation identifies the potential q(x) as A(0, x). Finally, Chapter 5 deals with two numerical approaches for solving an inverse spectral problem with a viscoelastic continuous spectral function. In the first approach, the A-equation is solved by reducing it to a system of Riccati equations using expansions in terms of shifted Chebyshev polynomials. In the second approach, the spectral function is approximated by stepwise spectral functions whose potentials, obtained using the Gelfand-Levitan method, serve as approximations for the underlying potential
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Scapellato, Andrea. "Integral operators and partial differential equations in Morrey type spaces." Doctoral thesis, Università di Catania, 2018. http://hdl.handle.net/10761/4043.

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Lo scopo di questa tesi è lo studio della limitatezza di alcuni operatori integrali in spazi funzionali di tipo Morrey. Inoltre si studia la regolarità di soluzioni di equazioni differenziali alle derivate parziali.
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25

Monguzzi, A. "ON THE REGULARITY OF SINGULAR INTEGRAL OPERATORS ON COMPLEX DOMAINS." Doctoral thesis, Università degli Studi di Milano, 2015. http://hdl.handle.net/2434/269875.

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Given a doman $\Omega$ in $C^n$, it is a classical problem to study the boundary behavior of functions which are holomorphic on $\Omega$. The boundary values of a given function are often expressed by means of singular integral operators. In this thesis we study this problem in two different settings with different motivations. In the first part we deal with a non-smooth version of the so-called worm domain in order to understand the role played by the pathological geometry of this domain. In the second part we study the problem in the case of a product Lipschitz surface and some boundedness results for biparameter singular integral operators are proved.
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Prats, Soler Martí. "Singular integral operators on sobolev spaces on domains and quasiconformal mappings." Doctoral thesis, Universitat Autònoma de Barcelona, 2015. http://hdl.handle.net/10803/314193.

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En aquesta tesi s’obtenen nous resultats sobre l’acotació d’operadors de Calderón-Zygmund en espais de Sobolev en dominis de Rd. En primer lloc es demostra un teorema de tipus T(P) vàlid per a Wn,p(U), a on U és un domini uniforme acotat de Rd, n és un nombre natural arbitrari, i p>d. Essencialment, el resultat obtingut afirma que un operador de Calderón-Zygmund de convolució és acotat en aquest espai si i solament si per a tot polinomi P de grau menor que n restringit al domini, T(P) pertany a Wn,p(U). Per a índexs p menors o iguals que d, es demostra una condició suficient per a l'acotació en termes de mesures de Carleson. En el cas n=1 i p<=d, es comprova que aquesta caracterització en termes de mesures de Carleson és també una condició necessària. El cas en què n és no enter i 02. La darrera aportació de la tesi és l'aplicació dels resultats anteriorment descrits a l'estudi de la regularitat de l'equació de Beltrami que satisfan les aplicacions quasiconformes. Essencialment, es demostra que si el coeficient de Beltrami pertany a l'espai Wn,p(U), essent U un domini Lipschitz del pla complex amb parametritzacions de la frontera en un cert espai de Besov i p>2, llavors l'aplicació quasiconforme associada està en l'espai Wn,p(U).
In this dissertation some new results on the boundedness of Calderón-Zygmund operators on Sobolev spaces on domains in Rd. First a T(P)-theorem is obtained which is valid for Wn,p (U), where U is a bounded uniform domain of Rd, n is a given natural number and p>d. Essentially, the result obtained states that a convolution Calderón-Zygmund operator is bounded on this function space if and only if T(P) belongs to Wn,p (U) for every polynomial P of degree smaller than n restricted to the domain. For indices p less or equal than d, a sufficient condition for the boundedness in terms of Carleson measures is obtained. In the particular case of n=1 and p<=d, this Carleson condition is shown to be necessary in fact. The case where n is not integer and 02. Finally, an application of the aforementioned results is given for quasiconformal mappings in the complex plane. In particular, it is checked that the regularity Wn,p(U) of the Beltrami coefficient of a quasiconformal mapping for a bounded Lipschitz domain U with boundary parameterizations in a certain Besov space and p>2, implies that the mapping itself is in Wn+1,p(U).
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Bishop, Shannon Renee Smith. "Gabor and wavelet analysis with applications to Schatten class integral operators." Diss., Georgia Institute of Technology, 2010. http://hdl.handle.net/1853/33976.

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This thesis addresses four topics in the area of applied harmonic analysis. First, we show that the affine densities of separable wavelet frames affect the frame properties. In particular, we describe a new relationship between the affine densities, frame bounds and weighted admissibility constants of the mother wavelets of pairs of separable wavelet frames. This result is also extended to wavelet frame sequences. Second, we consider affine pseudodifferential operators, generalizations of pseudodifferential operators that model wideband wireless communication channels. We find two classes of Banach spaces, characterized by wavelet and ridgelet transforms, so that inclusion of the kernel and symbol in appropriate spaces ensures the operator is Schatten p-class. Third, we examine the Schatten class properties of pseudodifferential operators. Using Gabor frame techniques, we show that if the kernel of a pseudodifferential operator lies in a particular mixed modulation space, then the operator is Schatten p-class. This result improves existing theorems and is sharp in the sense that larger mixed modulation spaces yield operators that are not Schatten class. The implications of this result for the Kohn-Nirenberg symbol of a pseudodifferential operator are also described. Lastly, Fourier integral operators are analyzed with Gabor frame techniques. We show that, given a certain smoothness in the phase function of a Fourier integral operator, the inclusion of the symbol in appropriate mixed modulation spaces is sufficient to guarantee that the operator is Schatten p-class.
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28

Oram, John A. "Best approximations from certain classes of functions defined by integral operators." Thesis, University of Newcastle Upon Tyne, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.386590.

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29

Hughes, Daniel Gordon John. "Spectral analysis of Dirac operators under integral conditions on the potential." Thesis, Cardiff University, 2012. http://orca.cf.ac.uk/43141/.

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We show that the absolutely continuous part of the spectral function of the one-dimensional Dirac operator on a half-line with a constant mass term and a real, square-integrable potential is strictly increasing throughout the essential spectrum (-∞,-1]U[1,∞). The proof is based on estimates for the transmission coefficient for the full-line scattering problem with a truncated potential and a subsequent limiting procedure for the spectral function. Furthermore, we show that the absolutely continuous spectrum persists when an angular momentum term is added, thus establishing the result for spherically symmetric Dirac operators in higher dimensions, too. Finally, with regard to this problem, we show that a sparse perturbation of a square integrable potential does not cause the absolutely continuous spectrum to become larger in the one-dimensional case. The final problem considered is regarding bound states, where we show that if the electric potential obeys the asymptotic bound C:=\lim sup_x→∞_ x|q(x)|<∞ then the eigenvalues outside of the spectral gap [-m,m] must obey Σ_n_(λ²_n-1)
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30

Ahmad, Khan Mumtaz, and Abukhammash Ghazi Salama. "A study on two variable analogues of certain fractional integral operators." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/97030.

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The Paper deals with a two variable analogues of certain fractional integral operators introduced by M. Saigo. Besides giving two variable analogues of earlier known fractional integral operators of one variable as special cases of newly defined operators, the paper establishes certain results in the form of theorems including integration by parts.
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31

Nixon, Steven Paul. "Theory and applications of the multiwavelets for compression of boundary integral operators." Thesis, University of Salford, 2004. http://usir.salford.ac.uk/2035/.

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In general the numerical solution of boundary integral equations leads to full coefficient matrices. The discrete system can be solved in O(N2) operations by iterative solvers of the Conjugate Gradient type. Therefore, we are interested in fast methods such as fast multipole and wavelets, that reduce the computational cost to O(N lnp N). In this thesis we are concerned with wavelet methods. They have proved to be very efficient and effective basis functions due to the fact that the coefficients of a wavelet expansion decay rapidly for a large class of functions. Due to the multiresolution property of wavelets they provide accurate local descriptions of functions efficiently. For example in the presence of corners and edges, the functions can still be approximated with a linear combination of just a few basis functions. Wavelets are attractive for the numerical solution of integral equations because their vanishing moments property leads to operator compression. However, to obtain wavelets with compact support and high order of vanishing moments, the length of the support increases as the order of the vanishingmoments increases. This causes difficulties with the practical use of wavelets particularly at edges and corners. However, with multiwavelets, an increase in the order of vanishing moments is obtained not by increasing the support but by increasing the number of mother wavelets. In chapter 2 we review the methods and techniques required for these reformulations, we also discuss how these boundary integral equations may be discretised by a boundary element method. In chapter 3, we discuss wavelet and multiwavelet bases. In chapter 4, we consider two boundary element methods, namely, the standard and non-standard Galerkin methods with multiwavelet basis functions. For both methods compression strategies are developed which only require the computation of the significant matrix elements. We show that they are O(N logp N) such significant elements. In chapters 5 and 6 we apply the standard and non-standard Galerkin methods to several test problems.
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32

Ballard, Grey M. "Asymptotic behavior of the eigenvalues of Toeplitz integral operators associated with the Hankel transform." Electronic thesis, 2008. http://dspace.zsr.wfu.edu/jspui/handle/10339/221.

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33

Zhao, Dandan, and 趙丹丹. "Integral inequalities and solvability of boundary value problems with p(t)-Laplacian operators." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2009. http://hub.hku.hk/bib/B42841823.

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34

Zhao, Dandan. "Integral inequalities and solvability of boundary value problems with p(t)-Laplacian operators." Click to view the E-thesis via HKUTO, 2009. http://sunzi.lib.hku.hk/hkuto/record/B42841823.

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35

Hocaoğlu, Ali Köksal. "Choquet integral based-morphological operators with applications to object detection and information fusion /." free to MU campus, to others for purchase, 2000. http://wwwlib.umi.com/cr/mo/fullcit?p9988671.

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36

Beaver, Scott. "Banach algebras of integral operators, off-diagonal decay, and applications in wireless communications /." For electronic version search Digital dissertations database. Restricted to UC campuses. Access is free to UC campus dissertations, 2004. http://uclibs.org/PID/11984.

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37

Awala, Hussein. "SINGULAR INTEGRAL OPERATORS ASSOCIATED WITH ELLIPTIC BOUNDARY VALUE PROBLEMS IN NON-SMOOTH DOMAINS." Diss., Temple University Libraries, 2017. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/453799.

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Mathematics
Ph.D.
Many boundary value problems of mathematical physics are modelled by elliptic differential operators L in a given domain Ω . An effective method for treating such problems is the method of layer potentials, whose essence resides in reducing matters to solving a boundary integral equation. This, in turn, requires inverting a singular integral operator, naturally associated with L and Ω, on appropriate function spaces on ƌΩ. When the operator L is of second order and the domain Ω is Lipschitz (i.e., Ω is locally the upper-graph of a Lipschitz function) the fundamental work of B. Dahlberg, C. Kenig, D. Jerison, E. Fabes, N. Rivière, G. Verchota, R. Brown, and many others, has opened the door for the development of a far-reaching theory in this setting, even though several very difficult questions still remain unanswered. In this dissertation, the goal is to solve a number of open questions regarding spectral properties of singular integral operators associated with second and higher-order elliptic boundary value problems in non-smooth domains. Among other spectral results, we establish symmetry properties of harmonic classical double layer potentials associated with the Laplacian in the class of Lipschitz domains in R2. An array of useful tools and techniques from Harmonic Analysis, Partial Differential Equations play a key role in our approach, and these are discussed as preliminary material in the thesis: --Mellin Transforms and Fourier Analysis; --Calderón-Zygmund Theory in Uniformly Rectifiable Domains; -- Boundary Integral Methods. Chapter four deals with proving invertibility properties of singular integral operators naturally associated with the mixed (Zaremba) problem for the Laplacian and the Lamé system in infinite sectors in two dimensions, when considering their action on the Lebesgue scale of p integrable functions, for 1 < p < ∞. Concretely, we consider the case in which a Dirichlet boundary condition is imposed on one ray of the sector, and a Neumann boundary condition is imposed on the other ray. In this geometric context, using Mellin transform techniques, we identify the set of critical integrability indexes p for which the invertibility of these operators fails. Furthermore, for the case of the Laplacian we establish an explicit characterization of the Lp spectrum of these operators for each p є (1,∞), as well as well-posedness results for the mixed problem. In chapter five, we study spectral properties of layer potentials associated with the biharmonic equation in infinite quadrants in two dimensions. A number of difficulties have to be dealt with, the most significant being the more complex nature of the singular integrals arising in this 4-th order setting (manifesting itself on the Mellin side by integral kernels exhibiting Mellin symbols involving hyper-geometric functions). Finally, chapter six, deals with spectral issues in Lipschitz domains in two dimensions. Here we are able to prove the symmetry of the spectra of the double layer potentials associated with the Laplacian. This is in essence a two-dimensional phenomenon, as known examples show the failure of symmetry in higher dimensions.
Temple University--Theses
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38

Blaimer, Bettina [Verfasser]. "Optimal Domain and Integral Extension of Operators Acting in Fréchet Function Spaces / Bettina Blaimer." Berlin : Logos Verlag, 2017. http://d-nb.info/1147809267/34.

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39

Strauss, Albrecht. "Integralformeln und a priori-Abschätzungen für das [delta bar]-Neumann-Problem." Bonn : [s.n.], 1988. http://catalog.hathitrust.org/api/volumes/oclc/18440543.html.

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40

Kowalski, Michael Władisław. "Comparative study of oscillatory integral, and sub-level set, operator norm estimates." Thesis, University of Edinburgh, 2010. http://hdl.handle.net/1842/4687.

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Oscillatory integral operators have been of interest to both mathematicians and physicists ever since the emergence of the work Theorie Analytique de la Chaleur of Joseph Fourier in 1822, in which his chief concern was to give a mathematical account of the diffusion of heat. For example, oscillatory integrals naturally arise when one studies the behaviour at infinity of the Fourier transform of a Borel measure that is supported on a certain hypersurface. One reduces the study of such a problem to that of having to obtain estimates on oscillatory integrals. However, sub-level set operators have only come to the fore at the end of the 20th Century, where it has been discovered that the decay rates of the oscillatory integral I(lambda) above may be obtainable once the measure of the associated sub-level sets are known. This discovery has been fully developed in a paper of A. Carbery, M. Christ and J.Wright. A principal goal of this thesis is to explore certain uniformity issues arising in the study of sub-level set estimates.
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41

Hein, Hoernig Ricardo Oliver. "Green's functions and integral equations for the Laplace and Helmholtz operators in impedance half-spaces." Phd thesis, Ecole Polytechnique X, 2010. http://pastel.archives-ouvertes.fr/pastel-00006172.

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Dans cette thèse on calcule la fonction de Green des équations de Laplace et Helmholtz en deux et trois dimensions dans un demi-espace avec une condition à la limite d'impédance. Pour les calculs on utilise une transformée de Fourier partielle, le principe d'absorption limite, et quelques fonctions spéciales de la physique mathématique. La fonction de Green est après utilisée pour résoudre numériquement un problème de propagation des ondes dans un demi-espace qui est perturbé de manière compacte, avec impédance, en employant des techniques des équations intégrales et la méthode d'éléments de frontière. La connaissance de son champ lointain permet d'énoncer convenablement la condition de radiation dont on a besoin. Des expressions pour le champ proche et lointain de la solution sont données, dont l'existence et l'unicité sont discutées brièvement. Pour chaque cas un problème benchmark est résolu numériquement. On expose étendument le fond physique et mathématique et on inclut aussi la théorie des problèmes de propagation des ondes dans l'espace plein qui est perturbé de manière compacte, avec impédance. Les techniques mathématiques développées ici sont appliquées ensuite au calcul de résonances dans un port maritime. De la même façon, ils sont appliqués au calcul de la fonction de Green pour l'équation de Laplace dans un demi-plan bidimensionnel avec une condition à la limite de dérivée oblique.
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42

Sant\'Anna, Douglas Azevedo. "Decaimento dos autovalores de operadores integrais gerados por séries de potências." Universidade de São Paulo, 2013. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-06052013-103553/.

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O principal objetivo deste trabalho e descrever o decaimento dos autovalores de operadores integrais gerados por núcleos definidos por séries de potências, mediante hipóteses sobre os coeficientes na série que representa o núcleo gerador. A análise e implementada em duas frentes: inicialmente, consideramos o caso em que o núcleo esta definido sobre a esfera unitária de \'R POT. m+1\', estendendo posteriormente a análise, para o caso da bola unitária do mesmo espaço. Em seguida, visando primordialmente o caso em que o núcleo esta definido sobre a esfera unitaria em \'C POT. m+1\', abordamos um caso mais geral, aquele no qual o núcleo esta definido por uma série de funções \'L POT. 2\'(X, u)-ortogonais, sendo (X, u) um espaço de medida arbitrário
The main target in this work is to deduce eigenvalue decay for integral operators generated by power series kernels, under general assumptions on the coefficients in the series representing the kernel. The analysis is twofold: firstly, we consider generating kernels defined on the unit sphere in \'R POT. m+1\', replacing the sphere with the unit ball in a subsequent stage. Secondly, we consider generating kernels defined on a general measure space (X, u) and possessing an \'L POT. 2\'(X, u)-orthogonal expansion there, an attempt to cover the case in which the kernel is defined on the unit sphere in \'C POT. m+1\'
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43

Aquino, Junielson Pantoja de. "Alguns resultados sobre a teoria de restrição da transformada de Fourier." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2016. http://hdl.handle.net/10183/159583.

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A análise harmônica e o ramo da matemática que estuda a representação de funções ou sinais como a sobreposição de ondas base. Ela investiga e generaliza as noções das séries de Fourier e da transformação de Fourier. Neste trabalho, investigou-se um teorema de restrição da transformada de Fourier devido a Mitsis e Mockenhaupt (uma generalização do teorema de Stein-Tomas). Foram realizados estudos analíticos sobre o método para operadores integrais oscilatórios, baseado na fase estacionária. Os resultados permitem deduzir o teorema de restrição no plano (em seu caso geral) e o teorema de Carleson-Sjölin.
Harmonic analysis is the mathematical branch that studies the function or signals representation as a base wave overlay. It investigates and generalizes the notions of Fourier series and of the Fourier transform. In this work, was investigated a restriction theorem of the Fourier transform due to Mitsis and Mockenhaupt (a generalization of Stein-Tomas theorem) . Were performed analytic studies on the method for oscillating integral operators, based in the stationary phase. The results allow deducing the restriction theorem on the plane (in the general case) and the Carleson-Sjölin theorem.
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44

Hanson-Hart, Zachary Aaron. "A Cauchy Problem with Singularity Along the Initial Hypersurface." Diss., Temple University Libraries, 2011. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/126171.

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Mathematics
Ph.D.
We solve a one-sided Cauchy problem with zero right hand side modulo smooth errors for the wave operator associated to a smooth symmetric 2-tensor which is Lorentz on the interior and degenerate at the boundary. The degeneracy of the metric at the boundary gives rise to singularities in the wave operator. The initial data prescribed at the boundary must be modified from the classical Cauchy problem to suit the problem at hand. The problem is posed on the interior and the local solution is constructed using microlocal analysis and the techniques of Fourier Integral Operators.
Temple University--Theses
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45

Ghassel, Ali. "The radon split of radially acting linear integral operators on h¦2 with uniformly bounded double norms." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/MQ48521.pdf.

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46

Silva, Nunes Ana Luisa. "Spectral approximation with matrices issued from discretized operators." Phd thesis, Université Jean Monnet - Saint-Etienne, 2012. http://tel.archives-ouvertes.fr/tel-00952977.

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In this thesis, we consider the numerical solution of a large eigenvalue problem in which the integral operator comes from a radiative transfer problem. It is considered the use of hierarchical matrices, an efficient data-sparse representation of matrices, especially useful for large dimensional problems. It consists on low-rank subblocks leading to low memory requirements as well as cheap computational costs. We discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and hence it is weakly singular. We access HLIB (Hierarchical matrices LIBrary) that provides, among others, routines for the construction of hierarchical matrix structures and arithmetic algorithms to perform approximative matrix operations. Moreover, it is incorporated the matrix-vector multiply routines from HLIB, as well as LU factorization for preconditioning, into SLEPc (Scalable Library for Eigenvalue Problem Computations) in order to exploit the available algorithms to solve eigenvalue problems. It is also developed analytical expressions for the approximate degenerate kernels and deducted error upper bounds for these approximations. The numerical results obtained with other approaches to solve the problem are used to compare with the ones obtained with this technique, illustrating the efficiency of the techniques developed and implemented in this work
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47

Ueda, Masaru. "The decomposition of the spaces of cusp forms of half-integral weight and trace formula of Hecke operators." 京都大学 (Kyoto University), 1987. http://hdl.handle.net/2433/86379.

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48

Bauer-Price, Pia. "The Selberg Trace Formula for PSL(2, OK) for imaginary quadratic number fields K of arbitrary class number." Bonn : [s.n.], 1991. http://catalog.hathitrust.org/api/volumes/oclc/26531368.html.

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49

Schoeman, Ilse Maria. "A theory of multiplier functions and sequences and its applications to Banach spaces / I.M. Schoeman." Thesis, North-West University, 2005. http://hdl.handle.net/10394/975.

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50

Li, Liangpan. "Local spectral asymptotics and heat kernel bounds for Dirac and Laplace operators." Thesis, Loughborough University, 2016. https://dspace.lboro.ac.uk/2134/23004.

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In this dissertation we study non-negative self-adjoint Laplace type operators acting on smooth sections of a vector bundle. First, we assume base manifolds are compact, boundaryless, and Riemannian. We start from the Fourier integral operator representation of half-wave operators, continue with spectral zeta functions, heat and resolvent trace asymptotic expansions, and end with the quantitative Wodzicki residue method. In particular, all of the asymptotic coefficients of the microlocalized spectral counting function can be explicitly given and clearly interpreted. With the auxiliary pseudo-differential operators ranging all smooth endomorphisms of the given bundle, we obtain certain asymptotic estimates about the integral kernel of heat operators. As applications, we study spectral asymptotics of Dirac type operators such as characterizing those for which the second coefficient vanishes. Next, we assume vector bundles are trivial and base manifolds are Euclidean domains, and study non-negative self-adjoint extensions of the Laplace operator which acts component-wise on compactly supported smooth functions. Using finite propagation speed estimates for wave equations and explicit Fourier Tauberian theorems obtained by Yuri Safarov, we establish the principle of not feeling the boundary estimates for the heat kernel of these operators. In particular, the implied constants are independent of self-adjoint extensions. As a by-product, we affirmatively answer a question about upper estimate for the Neumann heat kernel. Finally, we study some specific values of the spectral zeta function of two-dimensional Dirichlet Laplacians such as spectral determinant and Casimir energy. For numerical purposes we substantially improve the short-time Dirichlet heat trace asymptotics for polygons. This could be used to measure the spectral determinant and Casimir energy of polygons whenever the first several hundred or one thousand Dirichlet eigenvalues are known with high precision by other means.
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