The automorphism groups in the complex plane are defined, and we prove that they satisfy the group axioms. The automorphism group is derived for some domains. By applying the Riemann mapping theorem, it is proved that every automorphism group on simply connected domains that are proper subsets of the complex plane, is isomorphic to the automorphism group on the unit disc. / Automorfigrupperna i det komplexa talplanet definieras och vi bevisar att de uppfyller gruppaxiomen. Automorfigruppen på några domän härleds. Genom att applicera Riemanns avbildningssats bevisas att varje automorfigrupp på enkelt sammanhängande, öppna och äkta delmängder av det komplexa talplanet är isomorf med automorfigruppen på enhetsdisken.

Orthogonal polynomials, operators and commutation relations appear in many areas of mathematics, physics and engineering where they play a vital role. For instance, orthogonal functions in general are central to the development of Fourier series and wavelets which are essential to signal processing. In particular, as demonstrated in this thesis, orthogonal polynomials can be used to establish the L2-boundedness of singular integral operators which is a fundamental problem in harmonic analysis and a subject of extensive investigations. The Lp-convergence of Fourier series is closely related to the Lp-boundedness of singular integral operators. Many important relations in physical sciences are represented by operators satisfying various commutation relations. Such commutation relations play key roles in such areas as quantum mechanics, wavelet analysis, representation theory, spectral theory, and many others. This thesis consists of three main parts. The first part presents a new system of orthogonal polynomials, and establishes its relation to the previously studied systems in the class of Meixner–­Pollaczek polynomials. Boundedness properties of two singular integral operators of convolution type are investigated in the Hilbert spaces related to the relevant orthogonal polynomials. Orthogonal polynomials are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved in both cases that the two operators are bounded on L2-spaces, and estimates of the norms are obtained. The second part extends the investigation of the boundedness properties of the two singular integral operators to Lp-spaces on the real line, both in the weighted and unweighted spaces. It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by first proving boundedness for L2 and weak boundedness for L1, and then using interpolation to obtain boundedness for the intermediate spaces. To obtain boundedness for the remaining spaces, duality is used in the translation invariant case, while the weighted case is partly based on the methods developed by M. Riesz in his paper of 1928 for the conjugate function operator. The third and final part derives simple and explicit formulas for reordering elements in an algebra with three generators and Lie type relations. Centralizers and centers are computed as an example of an application of the formulas. / Ortogonala polynom, operatorer och kommutationsrelationer förekommer i många områden av matematik, fysik och teknik där de spelar en viktig roll. Till exempel ortogonala funktioner i allmänhet är centrala för utvecklingen av Fourierserier och wavelets som är väsentliga för signalbehandling. I synnerhet, såsom visats i denna avhandling, kan ortogonala polynom användas för att fastställa L2-begränsning av singulära integraloperatorer vilket är ett fundamentalt problem i harmonisk analys och föremål för omfattande forskning. Lp-konvergensen av Fourierserien är nära relaterad till Lp-begränsning av singulära integraloperatorer. Många viktiga relationer i fysik representeras av operatorer som uppfyller olika kommutationsrelationer. Sådana kommutationsrelationer spelar nyckelroller i områden som kvantmekanik, waveletanalys, representationsteori, spektralteori och många andra. Denna avhandling består av tre huvuddelar. Den första delen presenterar ett nytt system av ortogonala polynom, och etablerar dess förhållande till de tidigare studerade systemen i klassen Meixner–Pollaczek-polynom. Begränsningsegenskaper hos två singulära integraloperatorer av faltningstyp utreds i Hilbertrum relaterade till de relevanta ortogonala polynomen. Ortogonala polynom används för att bevisa begränsning i viktade rum och Fourieranalys används för att bevisa begränsning i det translationsinvarianta fallet. Det bevisas i båda fallen att de två operatorerna är begränsade på L2-rummen, och uppskattningar av normerna tas fram. Den andra delen utvidgar till Lp-rum på reella tallinjen undersökningen av begränsnings­egenskaperna hos de två singulära integral­operatorerna, både på viktade och oviktade rum. Det bevisas att de båda operatorerna är begränsade på dessa rum och uppskattningar av normerna erhålls. Detta uppnås genom att först bevisa begränsning för L2 och svag begränsning för L1, och sedan använda interpolation att erhålla begränsning för de mellanliggande rummen. För att erhålla begränsning för övriga Lp-rum används dualitet i det translationsinvarianta fallet, medan detta i det viktade fallet delvis bygger på en metod av M. Riesz i hans artikel från 1928 om konjugatfunktions­operatorn. Den tredje och sista delen härleder enkla och explicita formler för omkastning av element i en algebra med tre generatorer och relationer av Lie-typ. Som ett exempel på en tillämpning av formlerna beräknas centralisatorer och centra.

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This thesis consists of five papers (A-E), which examine the operation of infimal convolution and discuss its close connections to unilateral analysis, convex analysis, inequalities, approximation, and optimization. In particular, we attempt to provide a detailed investigation for both the convex and the non-convex case, including several examples. Paper (A) is both a survey of and a self-contained introduction to the operation of infimal convolution. In particular, we discuss the infimal value and minimizers of an infimal convolute, infimal convolution on subadditive functions, sufficient conditions for semicontinuity or continuity of an infimal convolute, "exactness," regularizing effects, continuity of the operation of infimal convolution, and approximation methods based on infimal convolution. A Young-type inequality, closely connected to the operation of infimal convolution, is studied in paper (B). The main results obtained are an equivalence theorem and a representation formula. In paper (C) we consider coercive, convex, proper, and lower sernicontinuous functions on a reflexive Banach space. For the infimal convolution of such functions we establish, in particular, different formulae. Moreover, we demonstrate the possibility of using the formulae obtained for solving special types of Hamilton-Jacobi equations. Furthermore, the operation of infimal convolution is interpreted from a physical viewpoint. Paper (D) presents properties of infimal convolution of functions that are uniformly continuous on bounded sets. In particular, we present regularization procedures by means of infimal convolution. The role of growth conditions on the functions under consideration is essential. Finally, in paper (E) we study semicontinuity, continuity, and differentiability of the infimal convolute of two convex functions. Moreover, under certain geometric conditions, the classical Moreau-Yosida approximation process is, roughly speaking, extended to the non-convex case. / Godkänd; 1994; 20070426 (ysko)

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<p>Godkänd; 1991; 20080410 (ysko)</p>

This thesis deals with applied mathematics with wavelets as a joint subject. There is an introduction and two extensive papers, of which one is already published in an international journal. The introduction presents wavelet theory including both the discrete and continuous wavelet transforms, the corresponding Fourier transforms, wavelet packets and the uncertainty principle. Moreover, it is a guide to applications. We consider applications that are strongly connected to the thesis and also other but more briefly. Also, the connection to both of the papers is included in the introduction. Paper 1 considers irregular sampling in shift-invariant spaces, such as for instance the spaces that are connected to a multiresolution analysis within wavelet theory. We set out the necessary theoretical aspects to enable reconstruction of an irregularly sampled function. Unlike most previous work in this area the method that is proposed in Paper 1 opens up for comparatively easy calculations of examples. Accordingly, we give a thorough exposition of one example of a sampling function. Paper 2 contains derivation and comparison of several different vibration analysis techniques for automatic detection of local defects in bearings. An extensive number of mathematical methods are suggested and evaluated through tests with both laboratory and industrial environment signals. Two out of the four best methods found are wavelet based, with an error rate of about 10%. Finally, there are many potentially performance improving additions included. / Godkänd; 2005; 20061108 (ysko)