This Licentiate Thesis consists of four chapters, which deal with a new Sobolev type function space called the space with multiweighted derivatives. This space is a generalization of the usual one dimensional Sobolev space. Chapter 1 is an introduction, where, in particular, the importance to study function spaces with weights is discussed and motivated. In Chapter 2 we consider and analyze some results of L. D. Kudryavtsev, where he investigated one dimensional Sobolev spaces. Moreover, in this chapter we present and prove analogous results by B. L. Baidel'dinov for generalized Sobolev spaces. These results are crucially for the proofs of the main results of this Licentiate Thesis. In Chapter 3 we prove some embedding theorems for these new generalized Sobolev spaces. The main results of Kudryavtsev and Baidel'dinov about characterization of the behavior of functions at a singularity take place in weak degeneration of spaces. However, with the help of our new embedding theorems we can extend these results to the case of strong degeneration. In Chapter 4 we prove some new estimates for each function in a Tchebychev system. In order to be able to study also compactness of the embeddings from Chapter 3 such estimates are crucial. I plan to study this question in detail in my further PhD studies. / <p>Godkänd; 2007; 20071107 (ysko)</p>

Homogenization is a mathematical theory for studying differential equations with rapidly oscillating coefficents. Many important problems in physics with one or several microscopic length scales give rise to this kind of equations. Hence there is a need for methods that enable an efficient treatment of such problems. To this end several homogenization techniques exist, ranging from the fairly abstract ones to those that are more oriented towards applications. This thesis is concerned with two such methods, namely the "asymptotic expansion method", also known as the "method of multiple scales", and multiscale convergence. The former method, sometimes referred to as the "engineering approach to homogenization" has, due to its versatility and intutive appeal, gained wide acceptance and popularity in the applied fields. However, it is not rigorous by mathematical standards. Multiscale convergence, introduced by Nguetseng in 1989, is a notion of weak convergence in Lp spaces that is designed to take oscillations into account. Although not the most general method around, multiscale convergence has become widely used by homogenizers because of its simplicity. In spite of its success, the multiscale theory is not yet sufficiently developed to be used in connection with certain nonlinear problems with several microscopic scales. In Paper A we extend some previously obtained results in multiscale convergence that enable us to homogenize a nonlinear problem with three scales. In Appendix to Paper A we present in more detail some results that were used in the proof of some of the main theorems in Paper A. Tribology is the science of bodies in relative motion interacting through a mechanical contact. An important aspect of tribology is to explain the principles of friction, lubrication and wear. Tribological phenomena are encountered everywhere in nature and technology and have a huge economical impact on society. An important example is that of two sliding solid surfaces interacting through a thin film of viscous fluid (lubricant). Hydrodynamic lubrication occurs when the pressure generated within the lubricant, through the viscosity of the fluid, is able to sustain an externally applied load. Many common bearings, e.g. journal bearings or slider bearings, operate according to this principle. As a branch of fluid dynamics, the mathematical foundations of lubrication theory are given by the Navier-Stokes equations, describing the motion of a viscous fluid. Because of the thin film assumption several simplifications are possible, leading to various reduced equations named after Osborne Reynolds, the founding father of lubrication theory. The Reynolds equation is used by engineers to compute the pressure distribution in various situations of thin film lubrication. For extremely thin films, it has been observed that the surface micro topography is an important factor in hydrodynamic performance. Hence it is important to understand the influence of surface roughness with small characteristic wavelength upon the pressure solution. Since the 1980s such problems have been increasingly studied by homogenization theory. The idea is to replace the original equation with a homogenized equation where the roughness effects are "averaged out". One problem consists of finding an algorithm that gives the homogenized equation. Another problem, consists of showing, by introducing the appropriate mathematical defintions, that the homogenized equation really is the correct one. Papers B and C investigate the effects of surface roughness by means of multiscale expansion of the pressure in various situations of hydrodynamic lubrication. Paper B, for which Paper A constitutes a rigorous basis, considers homogenization of the stationary Reynolds equation and roughness with two characteristic wavelengths. This leads to a multiscale problem and adds to the complexity of the homogenization process. To compare the homogenized solution to the solution of the unaveraged Reynolds equation, some numerical examples are also included. Paper C is devoted to homogenization of a variational principle which is a generalization of the unstationary Reynolds equation (both surfaces are rough). The advantage of adopting the calculus of variations viewpoint is that the recently introduced "variational bounds" can be computed. Bounds can be seen as a "cheap" alternative to computing the realtively costly homogenized solution. Several numerical examples are included to illustrate the utility of bounds. / Godkänd; 2008; 20080905 (ysko)

Derivatan är en fundamental del av matematiken. Denna uppsats kommer att handla om historiska framstegen inom matematiken som lett fram till att derivatan har kunnat definierats så som vi är vana att se den idag. / The derivative is a fundamental part of mathematics. This essay will be about historicaladvancements in mathematics, which led to the fact that the derivative has been definedas we are used to seeing it today.

Syftet med studien var att utveckla samt pröva ett analysverktyg för att se vilka matematiska förmågor fem olika matematiska digitala spel faktiskt erbjöd eleverna. Analysverktyget måste vara förenligt med kursplanen i matematik, för årskurs 1–3, och taluppfattning. Analysverktyget var baserat på Mathematical Competency Research Framework (MCRF) (Lithner et al., 2010). Vidare inkluderades relevanta förmågor från kursplanen i matematik och taluppfattning. En kodningsmanual samt ett kodningsschema användes i datainsamlingen. Kodningsmanualen innehåller 2–3 frågor formulerade från analysverktyget. Resultatet visar att analysverktyget behöver utveckla resonemangsförmågan samt kommunikationsförmågan. Två av de digitala spelen erbjöd begreppsförmågan, procedurförmågan samt kommunikationsförmågan. De andra tre erbjöd problemlösning, begreppsförmågan, procedurförmågan samt kommunikationsförmågan. För lärare innebär resultatet att analysverktyget kan användas för att undersöka vad de digitala spelen tillför undervisningen samt eleverna.

This thesis is devoted to some problems connected to the theory of homogenization of partial differential operators. The thesis consists of an elementary introduction and five different parts, A-E. In the introduction we give an elementary presentation of the basic ideas in the homogenization theory. Moreover, the introduction also serves as an overview of the field and points out where the results contained in this thesis fit in. The first part, Part A, consists of two papers with a complementary appendix and deals some purely theoretical parts of homogenization theory. Part B consists of one paper dealing with bounds of the homogenized operator. The third part, Part C, consists of two papers concerning some computational aspects of homogenization. Part D consists of two papers which show how the theoretical results from the homogenization theory can be practically used in composites engineering. Finally, Part E consists of two papers presented at international conferences, which consider some further mathematical and engineering aspects of the homogenization method. / Godkänd; 2002; 20061113 (haneit)

A very significant role in the estimation of different operators in analysis is played by the Hardy-Littlewood maximal function. There are a lot of papers dedicated to the study of properties of it, its variants, and their applications. One of the important variants of the Hardy-Littlewood maximal function is the so-called fractional maximal function, which is deeply connected to the Riesz potential operator. The main goal of the thesis is to establish analogues of some important properties of the Hardy-Littlewood maximal function for the fractional maximal function. In 1930 Hardy and Littlewood proved a remarkable result, known as the Hardy-Littlewood maximal theorem. Therefore its naturally arose a problem: what is an analogue of the Hardy-Littlewood maximal theorem for the fractional maximal function? In the thesis we will give an answer for this problem. Particularly, we will show that the so-called Hausdorff capacity and Morrey spaces, introduced by C. Morrey in 1938 in connection with some problems in elliptic partial differential equations and the theory of variations, naturally appears here. Moreover, recently Morrey spaces found important applications in connection with the Navier-Stokes and Schrödinger equations, elliptic problems with discontinuous coefficients and potential theory. The Hardy-Littlewood maximal theorem is deeply connected with the Stein-Wiener and Riesz-Herz equivalences. Analogues of these equivalences for the fractional maximal function are also given. In 1971 C. Fefferman and E. Stein, by using the Calderón-Zygmund decomposition, obtained the generalization of the maximal theorem of Hardy-Littlewood for a sequence of functions. This result of Fefferman and Stein found many important applications in Harmonic Analysis and its applications, e.g. in Signal Processing. In the thesis we will give an analogue of one part of the Fefferman-Stein maximal theorem for the fractional maximal operator. In 1952 A. Calderón and A. Zygmund published the paper "On Existence of Certain Singular Integrals", which has made a significant influence on the Analysis of the last 50 years. One of the main new tools used by A. Calderón and A. Zygmund was a special family of the decomposition of a given function in its "good" and "bad" parts. This decomposition provides a multidimensional substitution of the famous "sunrise" lemma by F. Riesz and it was used for proving a weak-type estimate for singular integrals. Furthermore, we want to emphasize that Calderón-Zygmund type decompositions have played an important and sometimes crucial role in the proofs of many fundamental results, such as the John-Nirenberg inequality, the theory of Ap-weights, Fefferman-Stein maximal theorem, etc. In the thesis it is showed that it is possible to construct an analogue of the Calderón-Zygmund decomposition for the Morrey spaces. / Godkänd; 2005; 20061004 (ysko)

Several signal processing applications today are based on the use of different transforms. The signals under consideration are written as a linear combination (or series) of some predefined set of functions. Traditionally, orthogonal bases have been used for this purpose, for example, in the discrete Fourier transform. The theory for orthogonal bases for Hilbert spaces can, however, be generalized to other sequences of functions, called frames. The first part of this thesis begins with an application-oriented introduction to the theory of frames and bases for separable Hilbert spaces. We explain similarities with and differences from the theory of orthogonal bases. Special attention is given to the relatively new theory of Gabor and Wavelet frames. We explain how they can be used for so-called time-frequency analysis. The main emphasis is on explaining fundamental similarities and differences between Gabor and wavelet frames. We also give an example of an application (OFDM) related to the second part of the thesis, for which nonorthogonal Gabor frames are superior to any orthogonal basis. The second part of this thesis concerns the current development of a standard for very high speed digital communication in ordinary telephone copper wires. It is the result of a cooperation with the Division of Signal Processing and Telia Research. We present a novel duplex method for Very high bitrate Digital Subscriber Lines (VDSL), called Zipper. It is intended to provide bit rates up to 52 Mbit per second, about 1000 times faster than the most common modems today. Zipper is based on Discrete Multi Tone (DMT) modulation. It uses an orthogonal basis of Gabor type for the signal transmission. Certain cyclical extensions are used to ensure the orthogonality between the basis functions. Zipper is proposed as a standard for VDSL to both the American National Standards Institute (ANSI) T1E1.4 group and in the European Telecommunication Standards Institute (ETSI) TM6 group. It will also be presented for the International Telecommunication Union (ITU). Telia Research is currently building a prototype together with ST Microelectronics (former SGS-Thomson), France. The first Zipper-VDSL modems are expected to be available on the mass market in the year 2001. The second part of this thext consists of a brief introduction to Zipper, an ANSI standard contribution and three conference papers. The standard contribution compares Zipper performance with competing standard proposals at that time: TDD and FDD. In the first conference paper we present a new and patented method for reducing the interference that the unshielded copper wires experience from radio transmissions. The two last conference papers present a low complexity method for reducing the so-called Peak to Average power Ratio (PAR) of the transmitted signal. PAR is a measure for the amount of rare but very high peaks in the signal. A reduced PAR allows for using a cheaper digital-to-analog converter and amplifier in the transmitter. / Godkänd; 2000; 20070318 (ysko)

This thesis consists of four papers and deals with supersingular rank one perturbations of self-adjoint operators and their models in Hilbert or Pontryagin spaces. Here, the term supersingular describes perturbation elements that are outside the underlying space but still obey a certain regularity conditions. The first two papers study certain Sturm-Liouville differential expressions that can be realised as Schrödinger operators. In Paper I we show that for the potential consisting of the inverse square plus a comparatively well-behaved term we can employ an existing model due to Kurasov to describe these operators in a Hilbert space. In particular, this approach is in good agreement with ODE techniques. In Paper II we study the inverse fourth power potential.While it is known that the ODE techniques still work, we show that the above model fails and thus that there are limits to the above operator theoretic approach. In Paper III we concentrate on generalising Kurasov's model. The original formulation assumes that the self-adjoint operator is semi-bounded, whereas we drop this requirement. We give two models with a Hilbert and Pontryagin space structure, respectively, and study the connections between the resulting constructions. Finally, in Paper IV, we consider the concrete case of the operator of multiplication by the independent variable, a self-adjoint operator whose spectrum covers the real line, and study its perturbations. This illustrates some of the formalism that was developed in the previous paper, and a number of more explicit results are obtained, especially regarding the spectra of the appearing perturbed operators. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.</p>

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.