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.

Musiikki ahdistuksen taitona : Filosofinen tutkimus musiikin eksistentiaalis-ontologisesta merkityksestä

Music as the Art of Anxiety: A Philosophical Approach to the Existential-Ontological Meaning of Music. The present research studies music as an art of anxiety from the points of view of both Martin Heidegger s thought and phenomenological philosophy in general. In the Heideggerian perspective, anxiety is understood as a fundamental mode of being (Grundbefindlichkeit) in human existence. Taken as an existential-ontological concept, anxiety is conceived philosophically and not psychologically. The central research questions are: what is the relationship between music and existential-ontological anxiety? In what way can music be considered as an art of anxiety? In thinking of music as a channel and manifestation of anxiety, what makes it a special case? What are the possible applications of phenomenology and Heideggerian thought in musicology? The main aim of the research is to develop a theory of music as an art of existential-ontological anxiety and to apply this theory to musicologically relevant phenomena. Furthermore, the research will contribute to contemporary musicological debates and research as it aims to outline the phenomenological study of music as a field of its own; the development of a specific methodology is implicit in these aims. The main subject of the study, a theory of music as an art of anxiety, integrates Heideggerian and phenomenological philosophies with critical and cultural theories concerning violence, social sacrifice, and mimetic desire (René Girard), music, noise and society (Jacques Attali), and the affect-based charme of music (Vladimir Jankélévitch). Thus, in addition to the subjective mood (Stimmung) of emptiness and meaninglessness, the philosophical concept of anxiety also refers to a state of disorder and chaos in general; for instance, to noise in the realm of sound and total (social) violence at the level of society. In this study, music is approached as conveying the existentially crucial human compulsion for signifying i.e., organizing chaos. In music, this happens primarily at the immediate level of experience, i.e. in affectivity, and also in relation to all of the aforementioned dimensions (sound, society, consciousness, and so on). Thus, music s existential-ontological meaning in human existence, Dasein, is in its ability to reveal different orders of existence as such. Indeed, this makes music the art of anxiety: more precisely, music can be existentially significant at the level of moods. The study proceeds from outlining the relevance of phenomenology and Heidegger s philosophy in musicology to the philosophical development of a theory of music as the art of anxiety. The theory is developed further through the study of three selected specific musical phenomena: the concept of a musical work, guitar smashing in the performance tradition of rock music, and Erik Bergman s orchestral work Colori ed improvvisazioni. The first example illustrates the level of individual human-subject in music as the art of anxiety, as a means of signifying chaos, while the second example focuses on the collective need to socio-culturally channel violence. The third example, being music-analytical, studies contemporary music s ability to mirror the structures of anxiety at the level of a specific musical text. The selected examples illustrate that, in addition to the philosophical orientation, the research also contributes to music analysis, popular music studies, and the cultural-critical study of music. Key words: music, anxiety, phenomenology, Martin Heidegger, ontology, guitar smashing, Erik Bergman, musical work, affectivity, Stimmung, René Girard

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.