Composition operators, Aleksandrov measures and value distribution of analytic maps in the unit disc

A composition operator is a linear operator that precomposes any given function with another function, which is held fixed and called the symbol of the composition operator. This dissertation studies such operators and questions related to their theory in the case when the functions to be composed are analytic in the unit disc of the complex plane. Thus the subject of the dissertation lies at the intersection of analytic function theory and operator theory. The work contains three research articles. The first article is concerned with the value distribution of analytic functions. In the literature there are two different conditions which characterize when a composition operator is compact on the Hardy spaces of the unit disc. One condition is in terms of the classical Nevanlinna counting function, defined inside the disc, and the other condition involves a family of certain measures called the Aleksandrov (or Clark) measures and supported on the boundary of the disc. The article explains the connection between these two approaches from a function-theoretic point of view. It is shown that the Aleksandrov measures can be interpreted as kinds of boundary limits of the Nevanlinna counting function as one approaches the boundary from within the disc. The other two articles investigate the compactness properties of the difference of two composition operators, which is beneficial for understanding the structure of the set of all composition operators. The second article considers this question on the Hardy and related spaces of the disc, and employs Aleksandrov measures as its main tool. The results obtained generalize those existing for the case of a single composition operator. However, there are some peculiarities which do not occur in the theory of a single operator. The third article studies the compactness of the difference operator on the Bloch and Lipschitz spaces, improving and extending results given in the previous literature. Moreover, in this connection one obtains a general result which characterizes the compactness and weak compactness of the difference of two weighted composition operators on certain weighted Hardy-type spaces.

Properties of Toeplitz operators on analytic function spaces : from function symbols to distributions

Toeplitz operators are among the most important classes of concrete operators with applications to several branches of pure and applied mathematics. This doctoral thesis deals with Toeplitz operators on analytic Bergman, Bloch and Fock spaces. Usually, a Toeplitz operator is a composition of multiplication by a function and a suitable projection. The present work deals with generalizing the notion to the case where the function is replaced by a distributional symbol. Fredholm theory for Toeplitz operators with matrix-valued symbols is also considered. The subject of this thesis belongs to the areas of complex analysis, functional analysis and operator theory. This work contains five research articles. The articles one, three and four deal with finding suitable distributional classes in Bergman, Fock and Bloch spaces, respectively. In each case the symbol class to be considered turns out to be a certain weighted Sobolev-type space of distributions. The Bergman space setting is the most straightforward. When dealing with Fock spaces, some difficulties arise due to unboundedness of the complex plane and the properties of the Gaussian measure in the definition. In the Bloch-type spaces an additional logarithmic weight must be introduced. Sufficient conditions for boundedness and compactness are derived. The article two contains a portion showing that under additional assumptions, the condition for Bergman spaces is also necessary. The fifth article deals with Fredholm theory for Toeplitz operators having matrix-valued symbols. The essential spectra and index theorems are obtained with the help of Hardy space factorization and the Berezin transform, for instance. The article two also has a part dealing with matrix-valued symbols in a non-reflexive Bergman space, in which case a condition on the oscillation of the symbol (a logarithmic VMO-condition) must be added.

Reconsidering Relativism and Intentionalism in Interpretation : Donald Davidson, Hermeneutics, and Pragmatism

This study focuses on two philosophical issues related to the interpretation of art. Firstly, it considers the role of authorial intentions in interpretation. Secondly, the study raises the issue of relativism in interpretation through a discussion of the relativistic tendencies apparent in the views of three major figures of contemporary philosophy: Joseph Margolis, Hans-Georg Gadamer, and Richard Rorty. The major goal of the thesis is to develop a theory of interpretation supporting the role of authorial intentions in interpretation on the basis of Donald Davidson s late philosophy of language and the holistic account of interpretation that underlies different parts of his philosophy. It is my belief that an intentionalist view of interpretation built on Davidsonian elements manages to form the most convincing defense of that interpretive position against the skepticism present in the views of Margolis, Gadamer, and Rorty. The theoretical issues addressed in the thesis are illuminated by discussions of case-examples, most importantly Richard Wagner s The Valkyrie, Thomas Adés America: A Prophecy, and some symphonies by Dimitri Shostakovich. In chapter one, I present a critical discussion of Margolis robust relativism. While finding Margolis criticism of the self-refutive argument plausible, I, nevertheless, argue that the relativistic logic Margolis offers should not be favored in interpretation. The first parts of chapter two outline Davidsonian intentionalism by presenting a reading of Davidson s later work in philosophy of language and mind, and by indicating its relationship to Davidson s views of literature. Then, I shall compare Davidson s ideas with some recent modest forms of intentionalism found in analytic aesthetics, and argue that Davidsonian intentionalism is in many respects more satisfactory compared to them. Chapter three engages Gadamer s hermeneutics by defending E.D. Hirsch s criticism of Gadamer. Uncovering the shortcomings in the replies of Gadamer s followers to Hirsch s criticism serves as a basis for the defense of intentionalism in interpretation carried out in the chapter. That defense is then extended with a discussion of some recent hermeneutic readings of Davidson s views. Chapter four deals with the standing of intentionalism through Rorty s pragmatist approach to literature. By indicating the position of pragmatist notions of aesthetic experience and imagination in Davidsonian intentionalism, it is shown that an intentionalist approach need not be as impoverished with regard to the value Rorty attributes to literature as he assumes. The concluding chapter outlines some ways in which one can be a pluralist with regard to art and interpretation without falling into relativism.

History, Transfer, Politics : Five studies on the legacy of Uppsala philosophy

The Uppsala school of Axel Hägerström can be said to have been the last genuinely Swedish philosophical movement. On the other hand, the Swedish analytic tradition is often said to have its roots in Hägerström s thought. This work examines the transformation from Uppsala philosophy to analytic philosophy from an actor-based historical perspective. The aim is to describe how a group of younger scholars (Ingemar Hedenius, Konrad Marc-Wogau, Anders Wedberg, Alf Ross, Herbert Tingsten, Gunnar Myrdal) colonised the legacy of Hägerström and Uppsala philosophy, and faced the challenges they met in trying to reconcile this legacy with the changing philosophical and political currents of the 1930s and 40s. Following Quentin Skinner, the texts are analysed as moves or speech acts in a particular historical context. The thesis consists of five previously published case studies and an introduction. The first study describes how the image of Hägerström as the father of the Swedish analytic tradition was created by a particular faction of younger Uppsala philosophers who (re-) presented the Hägerströmian philosophy as a parallel movement to logical empiricism. The second study examines the confrontations between Uppsala philosophy and logical empiricism in both the editorial board and in the pages of Sweden s leading philosophical journal Theoria. The third study focuses on how the younger generation redescribed Hägerströmian legal philosophical ideas (Scandinavian Legal Realism), while the fourth study discusses how they responded to the accusations of a connection between Hägerström s value nihilistic theory and totalitarianism. Finally, the fifth study examines how the Swedish social scientist and Social Democratic intellectual Gunnar Myrdal tried to reconcile value nihilism with a strong political programme for social reform. The contribution of this thesis to the field consists mainly in a re-evaluation of the role of Uppsala philosophy in the history of Swedish philosophy. From this perspective the Uppsala School was less a collection of certain definite philosophical ideas than an intellectual legacy that was the subject of fierce struggles. Its theories and ideas were redescribed in various ways by individual actors with different philosophical and political intentions.

Oireista puhuminen lääkärin vastaanotolla : Keskustelunanalyyttinen tutkimus lääkärin kysymyksistä

Talking about symptoms during medical consultation. A conversation analytical study of doctors questions This linguistically oriented conversation analytic study investigates doctors questions and patients answers during medical consultation. The focus is on 1) the syntactic constructions of the doctors questions concerning the patients symptoms, 2) the function of different types of syntactic constructions, and 3) the sequential placement of the questions. The data used in the study consist of 57 videotaped doctor patient encounters in Finnish primary health care. The study shows that the traditional division between open and closed questions is vague and needs to be examined further. Open wh-questions and closed yes/no questions form heterogeneous classes: some of the closed questions can be treated as open and vice versa. Wh-questions which occur during the physical examination are often constructed to elicit short answers. These questions can consist of one word (e.g. milloin when ) which does not move to a new topic but supports the unfinished activity of palpation. During the verbal examination, wh-questions are formulated to elicit long descriptions as answers. For example, by asking mites + X ( what about + X), the doctor can open up a new topic and simultaneously give the patient the opportunity to discuss the topic from his/her perspective. Almost half of the yes/no questions project longer than just a minimal answer (e.g. a short confirmation or rejection). In these questions, the doctors use verbal elements which show that more than just a minimal answer is required. They can, for example, add an indefinite element (joku some or mitään any ) to a yes/no question, add a conjunctive vai ( or ) to the end of the question and thus open a space for various types of answers, or add a suggested answer to the question. In addition, the results show that declarative questions not only check understanding, but display the doctor s diagnosing process, check whether the doctor can move on to the next topic or action, and display implicitly the doctor s idea of what is connected and what is relevant. One aim of the study is to describe how different syntactic structures work together. A typical question chain consists of two or three questions. The first question is an open wh-question that elicits a new topic and creates different types of presuppositions. Contingent questions are constructed as yes/no questions that seek an affirmative answer or as declarative sentences that seek confirmation. Contingent questions can function as repair initiators and thus support achieving mutual understanding. Therefore, they are tools for the doctor to construct a description of the medical problem collaboratively with the patient. The results add to the results of previous studies on questions in medical consultation, but also suggest some corrections. They provide additional evidence for the idea that different types of syntactic constructions are useful in different types of settings. However, they also show that the variety of questions that doctors use is more manifold and diverse than the variety introduced in earlier studies and textbooks.

Nonverbal predication in Erzya : Studies on morphosyntactic variation and part of speech distinctions

This dissertation consists of four articles and an introduction. The five parts address the same topic, nonverbal predication in Erzya, from different perspectives. The work is at the same time linguistic typology and Uralic studies. The findings based on a large corpus of empirical Erzya data, which was collected using several different methods and included recordings of the spoken language, made it possible for the present study to apply, then test and finally discuss the previous theories based on cross-linguistic data. Erzya makes use of multiple predication patterns which vary from totally analytic to the morphologically very complex. Nonverbal predicate clause types are classified on the basis of propositional acts in clauses denoting class-membership, identity, property and location. The predicates of these clauses are nouns, adjectives and locational expressions, respectively. The following three predication strategies in Erzya nonverbal predication can be identified: i. the zero-copula construction, ii. the predicative suffix construction and iii. the copula construction. It has been suggested that verbs and nouns cannot be clearly distinguished on morphological grounds when functioning as predicates in Erzya. This study shows that even though predicativity must not be considered a sufficient tool for defining parts of speech in any language, the Erzya lexical classes of adjective, noun and verb can be distinguished from each other also in predicate position. The relative frequency and degree of obligation for using the predicative suffix construction decreases when moving left to right on the scale verb adjective/locative noun ( identificational statement). The predicative suffix is the main pattern in the present tense over the whole domain of nonverbal predication in Standard Erzya, but if it is replaced it is most likely to be with a zero-copula construction in a nominal predication. This study exploits the theory of (a)symmetry for the first time in order to describe verbal vs. nonverbal predication. It is shown that the asymmetry of paradigms and constructions differentiates the lexical classes. Asymmetrical structures are motivated by functional level asymmetry. Variation in predication as such adds to the complexity of the grammar. When symmetric structures are employed, the functional complexity of grammar decreases, even though morphological complexity increases. The genre affects the employment of predication strategies in Erzya. There are differences in the relative frequency of the patterns, and some patterns are totally lacking from some of the data. The clearest difference is that the past tense predicative suffix construction occurs relatively frequently in Standard Erzya, while it occurs infrequently in the other data. Also, the predicative suffixes of the present tense are used more regularly in written Standard Erzya than in any other genre. The genre also affects the incidence of the translative in uľ(ń)ems copula constructions. In translations from Russian to Erzya the translative case is employed relatively frequently in comparison to other data. This study reveals differences between the two Mordvinic languages Erzya and Moksha. The predicative suffixes (bound person markers) of the present tense are used more regularly in Moksha in all kinds of nonverbal predicate clauses compared to Erzya. It should further be observed that identificational statements are encoded with a predicative suffix in Moksha, but seldom in Erzya. Erzya clauses are more frequently encoded using zero-constructions, displaying agreement in number only.

Vuorovaikutus ääniterapiassa : Keskustelunanalyyttinen tutkimus harjoittelun rakenteesta, terapeutin antamasta palautteesta ja oppimisesta

This conversation analytical study analyses the interactional practices adopted by speech therapists and their clients during their training in voice therapy. This study also describes how learning takes place during the therapy process. In contrast to traditional voice therapy studies, change is examined here by using qualitative research methodology, namely conversation analysis. This study describes the structures of interaction in voice therapy, shows how the shortcomings in the client s performance are evaluated and corrected and finally, how the voice training sequence and the participation changes during therapy. The database consists of 51 videotaped voice therapy sessions from six clients with voice disorders. The analytic focus is on the practices in one voice training exercise of the trilled /r/. All the sequences of this exercise (in total 36) and all adjacency pairs within (N = 627) were transcribed and analysed in detail. This study shows that voice training consists of successive model imitation adjacency pairs. This adjacency pair works as a resource in voice training. Furthermore, the use of this particular adjacency pair is an institutional practice in all therapies in this study. The structure of interaction in voice training sequences resembles the practices found in aphasia therapy and in speech therapy of children, as well as the practices of educational and counselling interaction and physiotherapy. More than half of the adjacency pairs were expanded to three (or more) part structures as client s responses were typically followed by therapist s feedback. With their feedback turns, therapists: 1) maintain training practice, 2) evaluate the problem of client s performance, 3) deliver information, 4) activate the client to observe the performance and 5) assist her in correcting the performance. This study describes the four different ways that therapists help their clients to improve the performance after encountering a problem. The longitudinal data shows that learning in therapy is manifested in the changing participation. As clients learn to identify their voice features, they can participate in evaluating or correcting their performances by themselves. This study describes the recurrent professional practices of voice therapists and shows how the institutional commitments of voice therapy are managed in and through talk and interaction. The study also provides detailed description of the management of help in voice training. By describing the interaction in training sequences, this study expands the conception of voice rehabilitation and how it can be researched. The results demonstrate that the learning process and therapy outcomes can be assessed by analysing interaction in therapy. Moreover, this analysis lays the foundation for a novel understanding of the practices in speech therapy and for the development of speech therapy theory. By revealing the activities of interaction, it also makes it possible to discuss them explicitly with speech therapy students. Key words: voice therapy, conversation analysis, institutional interaction, learning, change in participation, feedback, evaluation, error correction, self-repair

I konstens namn - scenen som arbetsmiljö : Arbetsplatsmobbning ur skådespelarperspektiv

Workplace bullying is a topic of current interest in Finland. Workplace bullying is found in all professions, including the artistic ones. This thesis aims to explore workplace bullying from the view of the Finland-Swedish actors as a phenomenon that within dramatic art is difficult to define due to the fact that the body and emotions of an actor constitute his or her working tools. The research aims to deepen the understanding of the actors’ working situation, and particularly of the difficulties and problems actors face when exercising their job. The research problems are: What forms of bullying are the actors exposed to? Who is bullying? How is the bullying received by the actors, and what are the possible consequences? The theoretical orientation of this thesis is based upon dialogical philosophy where phenomenology, hermeneutics and dialog meet in an orientation where the unseen is emphasized and made visible. Artistic leadership should be based upon a pedagogic understanding that by an open and equal dialog with the Other recognizes human diversity. The narrative research was undertaken by using an interview guide for the interviews with eleven actors, six women and five men with the voice of a sixth man represented by an article. The interviews, each on average 118 minutes, were recorded and transcribed. The method of discursive analysis was initiated by numerous reflective readings based on analytic induction. The inductive part of the analysis consisted of mapping out the individual experiences of bullying where after the process of finding connecting common features in the extensive material took place. The coded data was then deductively grouped together according to the research problems, and subgroups were formed for deeper description. The research findings show that workplace bullying is an everyday occurrence within the field of dramatic art. Actors are bullied by theatre managers and directors as well as by colleagues and other personnel. The main areas of bullying is depreciation of one’s professional skills, the existing jargon, sexual harassment, collective bullying and bullying because of personal qualities. A significant finding concerning this problem was the existing culture of silence. Even if actually seeing and hearing a colleague being bullied, few stood up to defend the person being bullied because of fear of retaliation. Even the person actually being the object for bullying found it difficult to take any actions.

Becoming a teacher: emerging teacher identity in mathematics teacher education

This research examines three aspects of becoming a teacher, teacher identity formation in mathematics teacher education: the cognitive and affective aspect, the image of an ideal teacher directing the developmental process, and as an on-going process. The formation of emerging teacher identity was approached in a social psychological framework, in which individual development takes place in social interaction with the context through various experiences. Formation of teacher identity is seen as a dynamic, on-going developmental process, in which an individual intentionally aspires after the ideal image of being a teacher by developing his/her own competence as a teacher. The starting-point was that it is possible to examine formation of teacher identity through conceptualisation of observations that the individual and others have about teacher identity in different situations. The research uses the qualitative case study approach to formation of emerging teacher identity, the individual developmental process and the socially constructed image of an ideal mathematics teacher. Two student cases, John and Mary, and the collective case of teacher educators representing socially shared views of becoming and being a mathematics teacher are presented. The development of each student was examined based on three semi-structured interviews supplemented with written products. The data-gathering took place during the 2005 2006 academic year. The collective case about the ideal image provided during the programme was composed of separate case displays of each teacher educator, which were mainly based on semi-structured interviews in spring term 2006. The intentions and aims set for students were of special interest in the interviews with teacher educators. The interview data was analysed following the modified idea of analytic induction. The formation of teacher identity is elaborated through three themes emerging from theoretical considerations and the cases. First, the profile of one s present state as a teacher may be scrutinised through separate affective and cognitive aspects associated with the teaching profession. The differences between individuals arise through dif-ferent emphasis on these aspects. Similarly, the socially constructed image of an ideal teacher may be profiled through a combination of aspects associated with the teaching profession. Second, the ideal image directing the individual developmental process is the level at which individual and social processes meet. Third, formation of teacher identity is about becoming a teacher both in the eyes of the individual self as well as of others in the context. It is a challenge in academic mathematics teacher education to support the various cognitive and affective aspects associated with being a teacher in a way that being a professional and further development could have a coherent starting-point that an individual can internalise.

Androgen receptor-dependent and independent functions of ARIP4

Androgen receptor (AR) is necessary for normal male phenotype development and essential for spermatogenesis. AR is a classical steroid receptor mediating actions of male sex steroids testosterone and 5-alpha-dihydrotestosterone. Numerous coregulators interact with the receptor and regulate AR activity on target genes. This study deals with the characterization of androgen receptor-interacting protein 4 (ARIP4). ARIP4 binds DNA, interacts with AR in vitro and in cultured yeast and mammalian cells, and modulates AR-dependent transactivation. ARIP4 is an active DNA-dependent ATPase, and this enzymatic activity is essential for the ability of ARIP4 to modulate AR function. On the basis of sequence homology in its ATPase domain, ARIP4 belongs to the SNF2 family of proteins involved in chromatin remodeling, DNA repair, and homologous recombination. Similar to its closest homologs ATRX and Rad54, ARIP4 does not seem to be a classical chromatin remodeling protein in that it does not appear to form large protein complexes in vivo or remodel mononucleosomes in vitro. However, ARIP4 is able to generate superhelical torsion on linear DNA fragments. ARIP4 is covalently modified by SUMO-1, and mutation of six potential SUMO attachment sites abolishes the ability of ARIP4 to bind DNA, hydrolyze ATP, and activate AR function. ARIP4 expression starts in early embryonic development. In mouse embryo ARIP4 is present mainly in the neural tube and limb buds. In adult mouse tissues ARIP4 expression is virtually ubiquitous. In mouse testis ARIP4 is expressed in the nuclei of Sertoli cells in a stage-dependent manner. ARIP4 is also present in the nuclei of Leydig cells, spermatogonia, pachytene and diplotene spermatocytes. Testicular expression pattern of ARIP4 does not differ significantly in wild-type, FSHRKO, and LuRKO mice. In the testis of hpg mice, ARIP4 is found mainly in interstitial cells and has very low, if any, expression in Sertoli and germ cells. Heterozygous Arip4+/ mice are fertile and appear normal; however, they are haploinsufficient with regard to androgen action in Sertoli cells. In contrast, Arip4 / embryos are not viable. They have significantly reduced body size at E9.5 and die by E11.5. Compared to wild-type littermates, Arip4 / embryos possess a higher percentage of apoptotic cells at E9.5 and E10.5. Fibroblasts derived from Arip4 / embryos cease growing after 2-3 passages and exhibit a significantly increased apoptosis and decreased proliferation rate than cells from wild-type embryos. Our findings demonstrate that ARIP4 plays an essential role in mouse embryonic development. In addition, testicular expression and AR coregulatory activity of ARIP4 suggest a role of ARIP4-AR interaction in the somatic cells of the testis.

Boundedness and convergence of singular integrals on fractal type sets

The topic of this dissertation lies in the intersection of harmonic analysis and fractal geometry. We particulary consider singular integrals in Euclidean spaces with respect to general measures, and we study how the geometric structure of the measures affects certain analytic properties of the operators. The thesis consists of three research articles and an overview. In the first article we construct singular integral operators on lower dimensional Sierpinski gaskets associated with homogeneous Calderón-Zygmund kernels. While these operators are bounded their principal values fail to exist almost everywhere. Conformal iterated function systems generate a broad range of fractal sets. In the second article we prove that many of these limit sets are porous in a very strong sense, by showing that they contain holes spread in every direction. In the following we connect these results with singular integrals. We exploit the fractal structure of these limit sets, in order to establish that singular integrals associated with very general kernels converge weakly. Boundedness questions consist a central topic of investigation in the theory of singular integrals. In the third article we study singular integrals of different measures. We prove a very general boundedness result in the case where the two underlying measures are separated by a Lipshitz graph. As a consequence we show that a certain weak convergence holds for a large class of singular integrals.

Composition operators on vector-valued BMOA and related function spaces

A composition operator is a linear operator between spaces of analytic or harmonic functions on the unit disk, which precomposes a function with a fixed self-map of the disk. A fundamental problem is to relate properties of a composition operator to the function-theoretic properties of the self-map. During the recent decades these operators have been very actively studied in connection with various function spaces. The study of composition operators lies in the intersection of two central fields of mathematical analysis; function theory and operator theory. This thesis consists of four research articles and an overview. In the first three articles the weak compactness of composition operators is studied on certain vector-valued function spaces. A vector-valued function takes its values in some complex Banach space. In the first and third article sufficient conditions are given for a composition operator to be weakly compact on different versions of vector-valued BMOA spaces. In the second article characterizations are given for the weak compactness of a composition operator on harmonic Hardy spaces and spaces of Cauchy transforms, provided the functions take values in a reflexive Banach space. Composition operators are also considered on certain weak versions of the above function spaces. In addition, the relationship of different vector-valued function spaces is analyzed. In the fourth article weighted composition operators are studied on the scalar-valued BMOA space and its subspace VMOA. A weighted composition operator is obtained by first applying a composition operator and then a pointwise multiplier. A complete characterization is given for the boundedness and compactness of a weighted composition operator on BMOA and VMOA. Moreover, the essential norm of a weighted composition operator on VMOA is estimated. These results generalize many previously known results about composition operators and pointwise multipliers on these spaces.

Renormalization methods in KAM theory

It is well known that an integrable (in the sense of Arnold-Jost) Hamiltonian system gives rise to quasi-periodic motion with trajectories running on invariant tori. These tori foliate the whole phase space. If we perturb an integrable system, the Kolmogorow-Arnold-Moser (KAM) theorem states that, provided some non-degeneracy condition and that the perturbation is sufficiently small, most of the invariant tori carrying quasi-periodic motion persist, getting only slightly deformed. The measure of the persisting invariant tori is large together with the inverse of the size of the perturbation. In the first part of the thesis we shall use a Renormalization Group (RG) scheme in order to prove the classical KAM result in the case of a non analytic perturbation (the latter will only be assumed to have continuous derivatives up to a sufficiently large order). We shall proceed by solving a sequence of problems in which theperturbations are analytic approximations of the original one. We will finally show that the approximate solutions will converge to a differentiable solution of our original problem. In the second part we will use an RG scheme using continuous scales, so that instead of solving an iterative equation as in the classical RG KAM, we will end up solving a partial differential equation. This will allow us to reduce the complications of treating a sequence of iterative equations to the use of the Banach fixed point theorem in a suitable Banach space.

Aspects of atomic decompositions and Bergman projections

The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.

Homoclinic Splitting without Trees

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, giving a simple construction of unstable KAM tori and their stable and unstable manifolds for analytic perturbations. When the coupling takes place through an even trigonometric polynomial in the angle variables, we extend analytically the solutions of the equations of motion, order by order in the perturbation parameter, to a large neighbourhood of the real line representing time. Subsequently, we devise an asymptotic expansion for the splitting (matrix) associated with a homoclinic point. This expansion consists of contributions that are manifestly exponentially small in the limit of vanishing gravity, by a shift-of-countour argument. Hence, we infer a similar upper bound for the splitting itself. In particular, the derivation of the result does not call for a tree expansion with explicit cancellation mechanisms.