On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Using Mathematical Morphology for Geometric Music Retrieval

The usual task in music information retrieval (MIR) is to find occurrences of a monophonic query pattern within a music database, which can contain both monophonic and polyphonic content. The so-called query-by-humming systems are a famous instance of content-based MIR. In such a system, the user's hummed query is converted into symbolic form to perform search operations in a similarly encoded database. The symbolic representation (e.g., textual, MIDI or vector data) is typically a quantized and simplified version of the sampled audio data, yielding to faster search algorithms and space requirements that can be met in real-life situations. In this thesis, we investigate geometric approaches to MIR. We first study some musicological properties often needed in MIR algorithms, and then give a literature review on traditional (e.g., string-matching-based) MIR algorithms and novel techniques based on geometry. We also introduce some concepts from digital image processing, namely the mathematical morphology, which we will use to develop and implement four algorithms for geometric music retrieval. The symbolic representation in the case of our algorithms is a binary 2-D image. We use various morphological pre- and post-processing operations on the query and the database images to perform template matching / pattern recognition for the images. The algorithms are basically extensions to classic image correlation and hit-or-miss transformation techniques used widely in template matching applications. They aim to be a future extension to the retrieval engine of C-BRAHMS, which is a research project of the Department of Computer Science at University of Helsinki.

Geometric Characterizations for Patterson-Sullivan Measures of Geometrically Finite Kleinian Groups

The main results of this thesis show that a Patterson-Sullivan measure of a non-elementary geometrically finite Kleinian group can always be characterized using geometric covering and packing constructions. This means that if the standard covering and packing constructions are modified in a suitable way, one can use either one of them to construct a geometric measure which is identical to the Patterson-Sullivan measure. The main results generalize and modify results of D. Sullivan which show that one can sometimes use the standard covering construction to construct a suitable geometric measure and sometimes the standard packing construction. Sullivan has shown also that neither or both of the standard constructions can be used to construct the geometric measure in some situations. The main modifications of the standard constructions are based on certain geometric properties of limit sets of Kleinian groups studied first by P. Tukia. These geometric properties describe how closely the limit set of a given Kleinian group resembles euclidean planes or spheres of varying dimension on small scales. The main idea is to express these geometric properties in a quantitative form which can be incorporated into the gauge functions used in the modified covering and packing constructions. Certain estimation results for general conformal measures of Kleinian groups play a crucial role in the proofs of the main results. These estimation results are generalizations and modifications of similar results considered, among others, by B. Stratmann, D. Sullivan, P. Tukia and S. Velani. The modified constructions are in general defined without reference to Kleinian groups, so they or their variants may prove useful in some other contexts in addition to that of Kleinian groups.

A Touch of Red : Archaeological and Ethnographic Approaches to Interpreting Finnish Rock Paintings

Approximately 125 prehistoric rock paintings have been found in the modern territory of Finland. The paintings were done with red ochre and are almost without exception located on steep lakeshore cliffs associated with ancient water routes. Most of the sites are found in the central and eastern parts of the country, especially on the shores of Lakes Päijänne and Saimaa. Using shore displacement chronology, the art has been dated to ca. 5000 – 1500 BC. It was thus created mainly during the Stone Age and can be associated with the so-called ‘Comb Ware’ cultures of the Subneolithic period. The range of motifs is rather limited, consisting mainly of schematic depictions of stick-figure humans, elks, boats, handprints and geometric signs. Few paintings include any evidence of narrative scenes, making their interpretation a rather difficult task. In Finnish archaeological literature, the paintings have traditionally been associated with ’sympathetic’ hunting magic, or the belief that the ritual shooting of the painted animals would increase hunting luck. Some writers have also suggested totemistic and shamanistic readings of the art. This dissertation is a critical review of the interpretations offered of Finnish rock art and an exploration of the potentials of archaeological and ethnographic research in increasing our knowledge of its meaning. Methods used include ’formal’ approaches such as archaeological excavation, landscape analysis and the application of neuropsychological research to the study of rock art, as well as ethnographically ’informed’ approaches that make use of Saami and Baltic Finnish ethnohistorical sources in interpretation. In conclusion, it is argued that although North European hunter-gatherer rock art is often thought to lie beyond the reach of ‘informed’ knowledge, the exceptional continuity of prehistoric settlement in Finland validates the informed approach in the interpretation of Finnish rock paintings. The art can be confidently associated with shamanism of the kind still practiced by the Saami of Northern Fennoscandia in the historical period. Evidence of similar shamanistic practices, concepts and cosmology are also found in traditional Finnish-Karelian epic poetry. Previous readings of the art based on ‘hunting magic’ and totemism are rejected. Most of the paintings appear to depict experiences of falling into a trance, of shamanic metamorphosis and trance journeys, and of ‘spirit helper’ beings comparable to those employed by the Saami shaman (noaidi). As demonstrated by the results of an excavation at the rock painting of Valkeisaari, the painted cliffs themselves find a close parallel in the Saami cult of the 'sieidi', or sacred cliffs and boulders worshipped as expressing a supernatural power. Like the Saami, the prehistoric inhabitants of the Finnish Lake Region seem to have believed that certain cliffs were ’alive’ and inhabited by the spirit helpers of the shaman. The rock paintings can thus be associated with shamanic vision quests, and the making of ‘art’ with an effort to socialize the other members of the community, especially the ritual specialists, with trance visions. However, the paintings were not merely to be looked at. The red ochre handprints pressed on images of elks, as well as the fact that many paintings appear ’smeared’, indicate that they were also to be touched – perhaps in order to tap into the supernatural potency inherent in the cliff and in the paintings of spirit animals.

Kohti lyyristä abstraktismia : Anitra Lucanderin 1950-luvun modernismi

Towards Lyrical Abstraction Anitra Lucander s Modernism in the 1950s Anitra Lucander (1918-2000) was one of the early pioneers of abstract art in Finland. During the Second World War Finnish art and cultural life was isolated and stagnated and figurative art was still dominant after the war. In the late 1940s and early 1950s, new international abstract art movements started to come to Finland. Anitra Lucander was one of the artists of the younger generation after the war who took an interest in the abstract movements in the early 1950s. At the beginning of the 1950s, abstract art came to Finland primarily in the form of Concretism, but simultaneously, a more delicate abstract movement emerged, and Anitra Lucander was among those cultivating such conceptions in her art. In this thesis, I observe and analyze through Anitra Lucander s art this central movement in Finnish modern art that has not yet been extensively studied. I examine how Anitra Lucander s art connects with the style change in Finnish art. I scrutinize the factors that affected Lucander and turned her towards abstract expression, and the effect her art had on emergence of abstract art in Finland. I will also consider the development of her art, the reception and critique of her art and the effect the critique had on her position in the 1950s art world. Because of a lack of earlier studies, I will undertake basic research, relying on empirical primary source material, where the starting point is to place the phenomenon under examination in the historical and cultural context. The most significant study materials are the artist s paintings and graphics from 1948 to 1960, newspaper and magazine articles from the same era, archive sources and interviews with Lucander s relatives, fellow artists and friends. An interesting aspect of the topic is the fact that Anitra Lucander was the only woman among the important pioneers of early Finnish abstract art. Through Lucander s art, I also examine the position of female artists in the tradition of Modernism as well as in the Finnish art world of the 1950s. This theoretical background is provided by the studies of feminist art historians, such as Marsha Meskimmon, Gill Perry, Griselda Pollock and Anne Middleton Wagner. Lucander s position in the male-dominated Finnish art scene of the 1950s, and how she achieved her position, emerges as one of the central themes of the study. I will also observe whether gender is evident in Lucander s art and expression, as well as her reception and critique compared to the reception of her male colleagues art. From a woman s point of view, I reveal the masculine rhetoric and gendered attitudes in the critique of the era. As a theoretical and methodological frame of reference, I use discourse analysis. Anitra Lucander encountered modernistic, international art movements during her journeys to Paris in the late 1940s and early 1950s. Her art evolved from the geometric Concretism of the early decade towards more delicate and painterly abstract expression. After the mid-1950s, she had developed her signature expression; through Cubism and a collage technique, she developed in her painting a delicate, coloristic imagery, which can be characterized as Lyrical Abstraction. Lucander did not consider abstract expression to be categorical, but saw the abstract and the nonfigurative as equals: the line between the abstract and the figurative is very often fleeting in her art. Already in her own time, Lucander achieved a position as one of the most talented young artists of her generation and her work was included in significant exhibitions. This success can definitely be attributed to the fact that she embraced Modernism in its extreme form, abstraction, already at the beginning of her career and networked with male painters who shared her outlook and modernistic expression. For her, this was either a conscious or an unconscious method of adapting to the male-dominated Finnish art field in the 1950s. In spite of acclaim and attention, Lucander had to encounter the gendered attitudes in the critique of the time, and her art was often perceived through stereotypical views as overly feminine and dependent. However, with her art, Lucander played an important role in the breakthrough for colorism and abstract art in Finland in the 1950s.

Half-occlusion processing in stereoscopic vision

This study investigated questions related to half-occlusion processing in human stereoscopic vision: (1) How does the depth location of a half-occluding figure affect the depth localization of adjacent monocular objects? (2) Is three-dimensional slant around vertical axis (geometric effect) affected by half-occlusion constraints? and (3) How the half-occlusion constraints and surface formation processes are manifested in stereoscopic capture? Our results showed that the depth localization of binocular objects affects the depth localization of discrete monocular objects. We also showed that the visual system has a preference for a frontoparallel surface interpretation if the half-occlusion configuration allows multiple interpretation alternatives. When the surface formation was constrained by textures, our results showed that a process of rematching spreading determines the resulting perception and that the spreading can be limited by illusory contours that support the presence of binocularly unmatched figures. The unmatched figures could be present, if the inducing figures producing the illusory surface contained binocular image differences that provided cues for quantitative da Vinci stereopsis. These findings provide evidence of the significant role of half-occlusions in stereoscopic processing.

Organic contaminants - occurrence and biological effects in the Baltic Sea

The Baltic Sea was studied with respect to selected organic contaminants and their ecotoxicology. The research consisted of analyses of total hydrocarbons, polycyclic aromatic hydrocarbons, bile metabolites, hepatic ethoxyresorufin-O-deethylase (EROD) activity, polychlorinated biphenyls (PCBs) and organochlorine pesticides (OCPs). The contaminants were measured from various matrices, such as seawater, sediment and biota. The methods of analysis were evaluated and refined to comparability of the results. Polyaromatic hydrocarbons, originating from petroleum, are known to be among the most harmful substances to the marine environment. In Baltic subsurface water, seasonal dependence of the total hydrocarbon concentrations (THCs) was seen. Although concentrations of parent polycyclic aromatic hydrocarbons (PAHs) in sediment surface varied between 64 and 5161 ug kg-1 (dw), concentrations above 860 ug kg-1 (dw) were found in all the studied sub-basins of the Baltic Sea. Concentrations commonly considered to substantially increase the risk of liver disease and reproductive impairment in fish, as well as potential effects on growth (above 1000 ug kg-1 dw), were found in all the studied sub-basins of the Baltic Sea except Kattegat. Thus, considerable pollution in sediments was indicated. In bivalves, the sums of 12 PAHs varied on a wet weight basis between 44 and 298 ug kg-1 (ww). The predominant PAHs were high molecular weight and the PAH profiles of M. balthica differed from those found in sediment from the same area. The PAHs were both pyrolytic and petrogenic in origin, and a contribution from diesel engines was found, which indicates pollution of the Baltic Sea, most likely caused by the steadily increasing shipping in the area. The HPLC methods developed for hepatic EROD activity and bile metabolite measurements proved to be fast and suitable for the study of biological effects. A mixed function oxygenase enzyme system in Baltic Sea perch collected from the Gulf of Finland was induced slightly: EROD activity in perch varied from 0.30 14 pmol min-1 mg-1 protein. This range can be considered to be comparable to background values. Recent PAH exposure was also indicated by enhanced levels (213 and 1149 ug kg-1) of the bile metabolite 1-hydroxypyrene. No correlation was indicated between hepatic EROD activity and concentration of 1-hydroxypyrene in bile. PCBs and OCPs were observed in Baltic Sea sediment, bivalves and herring. Sums of seven CBs in surface sediment (0 5 cm) ranged from 0.04 to 6.2 ug kg-1 (dw) and sums of three DDTs from 0.13 to 5.0 ug kg-1 (dw). The highest levels of contaminants were found in the most eastern area of the Gulf of Finland where the highest total carbon and nitrogen content was found and where the lowest percentage proportion of p,p -DDT was found. The highest concentrations of CBs and the lowest concentration of DDTs were found in M. balthica from the Gulf of Finland. The highest levels of DDTs were found in M. balthica from the Hanö Bight, which is the outer part of the Bornholm Basin close to the Swedish mainland. In bivalves, the sums of seven CBs were 72 108 ug kg-1 (lw) and the sums of three DDTs were 66 139 ug kg-1 (lw). Results from temporal trend monitoring showed, that during the period 1985 2002, the concentrations of seven CBs in two-year-old female Baltic herring were clearly decreased, from 9 16 to 2 6 ug kg-1 (ww) in the northern Baltic Sea. At the same time, concentrations of three DDTs declined from 8 15 to 1 5 ug kg-1 (ww). The total concentration of the fat-soluble CBs and DDTs in Baltic herring muscle was shown to be age-dependent; the average concentrations in ten-year-old Baltic herring were three to five-fold higher than in two-year-old herring. In Baltic herring and bivalves, as well as in surface sediments, CB 138 and CB153 were predominant among CBs, whereas among DDTs p,p'-DDD predominated in sediment and p,p'-DDE in bivalves and Baltic herring muscle. Baltic Sea sediments are potential sources of contaminants that may become available for bioaccumulation. Based on ecotoxicological assessment criteria, cause for concern regarding CBs in sediments was indicated for the Gulf of Finland and the northern Baltic Proper, and for the northern Baltic Sea regarding CBs in Baltic herring more than two years old. Statistical classification of selected organic contaminants indicated high-level contamination for p,p'-DDT, p,p'-DDD, p,p'-DDE, total DDTs, HCB, CB118 and CB153 in muscle of Baltic herring in age groups two to ten years; in contrast, concentrations of a-HCH and g-HCH were found to be moderate. The concentrations of DDTs and CBs in bivalves is sufficient to cause biological effects, and demonstrates that long-term biological effects are still possible in the case of DDTs in the Hanö Bight.

Autoxidation of conjugated linoleic acid methyl ester in the presence of α-tocopherol: the hydroperoxide pathway

The autoxidation of conjugated linoleic acid (CLA) is poorly understood in spite of increasing interest in the beneficial biological properties of CLA and growing consumption of CLA-rich foods. In this thesis, the autoxidation reactions of the two major CLA isomers, 9-cis,11-trans-octadecadienoic acid and 10-trans,12-cis-octadecadienoic acid, are investigated. The results contribute to an understanding of the early stages of the autoxidation of CLA methyl ester, and provide for the first time a means of producing and separating intact CLA methyl ester hydroperoxides as well as basic knowledge on lipid hydroperoxides and their hydroxy derivatives. Conjugated diene allylic monohydroperoxides were discovered as primary autoxidation products formed during autoxidation of CLA methyl esters in the presence and absence of α-tocopherol. This established that one of the autoxidation pathways of CLA methyl ester is the hydroperoxide pathway. Hydroperoxides were produced from the two major CLA methyl esters by taking advantage of the effect of α-tocopherol to promote hydroperoxide formation. The hydroperoxides were analysed and separated first as methyl hydroxyoctadecadienoates and then as intact hydroperoxides by HPLC. The isolated products were characterized by UV, GC-MS, and NMR techniques. In the presence of a high amount of α-tocopherol, the autoxidation of CLA methyl ester yields six kinetically-controlled conjugated diene monohydroperoxides and is diastereoselective in favour of one particular geometric isomer as a pair of enantiomers. The primary autoxidation products produced from the two major CLA isomers include new positional isomers of conjugated diene monohydroperoxides, the 8-, 10-, 12-, and 14-hydroperoxyoctadecadienoates. Furthermore, two of these new positional isomers have an unusual structure for a cis,trans lipid hydroperoxide where the allylic methine carbon is adjacent to the cis instead of the usual trans double bond. The 1H and 13C NMR spectra of nine isomeric methyl hydroxyoctadecadienoates and of ten isomeric methyl hydroperoxyoctadecadienoates including the unusual cis,trans hydroperoxides, i.e. Me 8-OOH-9c,11t and Me 14-OOH-10t,12c, were fully assigned with the aid of 2D NMR spectroscopy. The assigned NMR data enabled determination of the effects of the hydroxyl and hydroperoxyl groups on the carbon chemical shifts of CLA isomers, identification of diagnostic signals, and determination of chemical shift differences of the olefinic resonances that may help with the assignment of structure to as yet unknown lipid hydroperoxides either as hydroxy derivatives or as intact hydroperoxides. A mechanism for the hydroperoxide pathway of CLA autoxidation in the presence of a high amount of α-tocopherol was proposed based on the characterized primary products, their relative distribution, and theoretical calculations. This is an important step forward in CLA research, where exact mechanisms for the autoxidation of CLA have not been presented before. Knowledge of these hydroperoxide formation steps is of crucial importance for understanding the subsequent steps and the different pathways of the autoxidation of CLA. Moreover, a deeper understanding of the autoxidation mechanisms is required for ensuring the safety of CLA-rich foods. Knowledge of CLA oxidation and how it differs from the oxidation of nonconjugated polyunsaturated fatty acids may also be the key to understanding the biological mechanisms of CLA activity.

Cavity-enhanced laser spectroscopic studies of vibrational overtones of acetylene

This thesis contains five experimental spectroscopic studies that probe the vibration-rotation energy level structure of acetylene and some of its isotopologues. The emphasis is on the development of laser spectroscopic methods for high-resolution molecular spectroscopy. Three of the experiments use cavity ringdown spectroscopy. One is a standard setup that employs a non-frequency stabilised continuous wave laser as a source. In the other two experiments, the same laser is actively frequency stabilised to the ringdown cavity. This development allows for increased repetition rate of the experimental signal and thus the spectroscopic sensitivity of the method is improved. These setups are applied to the recording of several vibration-rotation overtone bands of both H(12)C(12)CH and H(13)C(13)CH. An intra-cavity laser absorption spectroscopy setup that uses a commercial continuous wave ring laser and a Fourier transform interferometer is presented. The configuration of the laser is found to be sub-optimal for high-sensitivity work but the spectroscopic results are good and show the viability of this type of approach. Several ro-vibrational bands of carbon-13 substituted acetylenes are recorded and analysed. Compared with earlier work, the signal-to-noise ratio of a laser-induced dispersed infrared fluorescence experiment is enhanced by more than one order of magnitude by exploiting the geometric characteristics of the setup. The higher sensitivity of the spectrometer leads to the observation of two new symmetric vibrational states of H(12)C(12)CH. The precision of the spectroscopic parameters of some previously published symmetric states is also improved. An interesting collisional energy transfer process is observed for the excited vibrational states and this phenomenon is explained by a simple step-down model.

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.

Convex-transitive Banach spaces and their hyperplanes

The topic of this dissertation is the geometric and isometric theory of Banach spaces. This work is motivated by the known Banach-Mazur rotation problem, which asks whether each transitive separable Banach space is isometrically a Hilbert space. A Banach space X is said to be transitive if the isometry group of X acts transitively on the unit sphere of X. In fact, some weaker symmetry conditions than transitivity are studied in the dissertation. One such condition is an almost isometric version of transitivity. Another investigated condition is convex-transitivity, which requires that the closed convex hull of the orbit of any point of the unit sphere under the rotation group is the whole unit ball. Following the tradition developed around the rotation problem, some contemporary problems are studied. Namely, we attempt to characterize Hilbert spaces by using convex-transitivity together with the existence of a 1-dimensional bicontractive projection on the space, and some mild geometric assumptions. The convex-transitivity of some vector-valued function spaces is studied as well. The thesis also touches convex-transitivity of Banach lattices and resembling geometric cases.

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.