This thesis consists of three papers, all of them on the topic of function spaces on fractals. The papers summarised in this thesis are: Paper I Mats Bodin, Wavelets and function spaces on Mauldin-Williams fractals, Research Report in Mathematics No. 7, Umeå University, 2005. Paper II Mats Bodin, Harmonic functions and Lipschitz spaces on the Sierpinski gasket, Research Report in Mathematics No. 8, Umeå University, 2005. Paper III Mats Bodin, A discrete characterisation of Lipschitz spaces on fractals, Manuscript. The first paper deals with piecewise continuous wavelets of higher order in Besov spaces defined on fractals. A. Jonsson has constructed wavelets of higher order on fractals, and characterises Besov spaces on totally disconnected self-similar sets, by means of the magnitude of the coefficients in the wavelet expansion of the function. For a class of fractals, W. Jin shows that such wavelets can be constructed by recursively calculating moments. We extend their results to a class of graph directed self-similar fractals, introduced by R. D. Mauldin and S. C. Williams. In the second paper we compare differently defined function spaces on the Sierpinski gasket. R. S. Strichartz proposes a discrete definition of Besov spaces of continuous functions on self-similar fractals having a regular harmonic structure. We identify some of them with Lipschitz spaces introduced by A. Jonsson, when the underlying domain is the Sierpinski gasket. We also characterise some of these spaces by means of the magnitude of the coefficients of the expansion of a function in a continuous piecewise harmonic base. The last paper gives a discrete characterisation of certain Lipschitz spaces on a class of fractal sets. A. Kamont has discretely characterised Besov spaces on intervals. We give a discrete characterisation of Lipschitz spaces on fractals admitting a type of regular sequence of triangulations, and for a class of post critically finite self-similar sets. This shows that, on some fractals, certain discretely defined Besov spaces, introduced by R. Strichartz, coincide with Lipschitz spaces introduced by A. Jonsson and H. Wallin for low order of smoothness.

This thesis is devoted to the study of different generalizations of the classical conception of a function of bounded variation. First, we study the functions of bounded p-variation introduced by Wiener in 1924. We obtain estimates of the total p-variation (1<p<∞) and other related functionals for a periodic function f in Lp([0,1]) in terms of its Lp-modulus of continuity ω(f;δ)p. These estimates are sharp for any rate of decay of ω(f;δ)p. Moreover, the constant coefficients in them depend on parameters in an optimal way. Inspired by these results, we consider the relationship between the Riesz type generalized variation vp,α(f) (1<p<∞, 0≤α≤1-1/p) and the modulus of p-continuity  ω1-1/p(f;δ). These functionals generate scales of spaces that connect the space of functions of bounded p-variation and the Sobolev space Wp1. We prove sharp estimates of vp,α(f) in terms of ω1-1/p(f;δ). In the same direction, we study relations between moduli of p-continuity and q-continuity for 1<p<q<∞. We prove an inequality that estimates ω1-1/p(f;δ) in terms of ω1-1/q(f;δ). The inequality is sharp for any order of decay of ω1-1/q(f;δ). Next, we study another generalization of bounded variation: the so-called bounded Λ-variation, introduced by Waterman in 1972. We investigate relations between the space ΛBV of functions of bounded Λ-variation, and classes of functions defined via integral smoothness properties. In particular, we obtain the necessary and sufficient condition for the embedding of the class Lip(α;p) into ΛBV. This solves a problem of Wang (2009). We consider also functions of two variables. Applying our one-dimensional result, we obtain sharp estimates of the Hardy-Vitali type p-variation of a bivariate function in terms of its mixed modulus of continuity in Lp([0,1]2). Further, we investigate Fubini-type properties of the space Hp(2) of functions of bounded Hardy-Vitali p-variation. This leads us to consider the symmetric mixed norm space Vp[Vp]sym of functions of bounded iterated p-variation. For p>1, we prove that Hp(2) is not embedded into Vp[Vp]sym, and that Vp[Vp]sym is not embedded into Hp(2). In other words, Fubini-type properties completely fail in the class of functions of bounded Hardy-Vitali type p-variation for p>1. / Baksidestext The classical concept of the total variation of a function has been extended in several directions. Such extensions find many applications in different areas of mathematics. Consequently, the study of notions of generalized bounded variation forms an important direction in the field of mathematical analysis. This thesis is devoted to the investigation of various properties of functions of generalized bounded variation.  In particular, we obtain the following results: sharp relations between spaces of generalized bounded variation and spaces of functions  defined by integral smoothness conditions  (e.g., Sobolev and Besov spaces); optimal properties of certain scales of function spaces of frac- tional smoothness generated by functionals of variational type; sharp embeddings within  the scale of spaces of functions of bounded p-variation; results concerning bivariate functions of bounded p-variation, in particular sharp estimates of total variation in terms of the mixed Lp-modulus of continuity, and Fubini-type properties.

This thesis consists of three chapters. The first chapter focuses on adaptive smoothing splines for fitting functions with varying roughness. In the first part of the first chapter, we study an asymptotically optimal procedure to choose the value of a discretized version of the variable smoothing parameter in adaptive smoothing splines. With the choice given by the multivariate version of the generalized cross validation, the resulting adaptive smoothing spline estimator is shown to be consistent and asymptotically optimal under some general conditions. In the second part, we derive the asymptotically optimal local penalty function, which is subsequently used for the derivation of the locally optimal smoothing spline estimator. In the second chapter, we propose a Lipschitz regularity based statistical model, and apply it to coordinate measuring machine (CMM) data to estimate the form error of a manufactured product and to determine the optimal sampling positions of CMM measurements. Our proposed wavelet-based model takes advantage of the fact that the Lipschitz regularity holds for the CMM data. The third chapter focuses on the classification of functional data which are known to be well separable within a particular interval. We propose an interval based classifier. We first estimate a baseline of each class via convex optimization, and then identify an optimal interval that maximizes the difference among the baselines. Our interval based classifier is constructed based on the identified optimal interval. The derived classifier can be implemented via a low-order-of-complexity algorithm.

Thesis (Ph.D.)--Flinders University, School of Informatics and Engineering, Dept. of Mathematics. / Typescript bound. Includes bibliographical references: (leaves 135-140) and index. Also available online.

In this work we analyze some selection principles over some classes of hyperspaces. In the first part we consider selective variations of tightness over a class of function spaces whose topologies are determined by bornologies on the space. As results, we extend several well known translations between covering properties and closure properties of the topology of pointwise convergence. In the second part we consider artificial hyperspaces that assist the analysis of productive topological properties. We emphasize the results characterizing productively ccc preorders and the characterization of the Lindelöf property via closed projections. / Neste trabalho analisamos alguns princípios seletivos quando considerados sobre alguns tipos de hiperespaços. Na primeira parte consideramos variações seletivas do tightness sobre diversos tipos de espaços de funções, cujas topologias são determinadas por bornologias no espaço. Como resultados, estendemos diversas traduções conhecidas entre propriedades de recobrimento e propriedades de convergência na topologia da convergência pontual. Na segunda parte consideramos hiperespaços artificiais que auxiliam na análise de propriedades topológicas produtivas. Destacamos os resultados que caracterizam as pré-ordens produtivamente ccc e a caracterização da propriedade de Lindelöf em termos de projeções fechadas.

If T:C(H,X)-->Y is a bounded linear operator then there exists a unique weakly regular finitely additive set function m:-->L(X,Y**) so that T(f) = ∫Hfdm. In this paper, bounded linear operators on C(H,X) are studied in terms the measure given by this representation theorem. The first chapter provides a brief history of representation theorems of these classes of operators. In the second chapter the represenation theorem used in the remainder of the paper is presented. If T is a weakly compact operator on C(H,X) with representing measure m, then m(A) is a weakly compact operator for every Borel set A. Furthermore, m is strongly bounded. Analogous statements may be made for many interesting classes of operators. In chapter III, two classes of operators, weakly precompact and QSP, are studied. Examples are provided to show that if T is weakly precompact (QSP) then m(A) need not be weakly precompact (QSP), for every Borel set A. In addition, it will be shown that weakly precompact and GSP operators need not have strongly bounded representing measures. Sufficient conditions are provided which guarantee that a weakly precompact (QSP) operator has weakly precompact (QSP) values. A sufficient condition for a weakly precomact operator to be strongly bounded is given. In chapter IV, weakly precompact subsets of L1(μ,X) are examined. For a Banach space X whose dual has the Radon-Nikodym property, it is shown that the weakly precompact subsets of L1(μ,X) are exactly the uniformly integrable subsets of L1(μ,X). Furthermore, it is shown that this characterization does not hold in Banach spaces X for which X* does not have the weak Radon-Nikodym property.

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This work is dealing with almost-compact embeddings of function spaces, in particular, the class of classical and weak Lorentz spaces with a norm given by a general weight fuction is studied. These spaces are not Banach function spaces in general, thus the almost-compact em- bedding is defined for more general sturctures of rearrangement-invariant lattices. A general characterization of when an r.i. lattice is almost-compactly embedded into a Lorentz space, involving an optimal constant of a certain continuous embedding, is proved. Based on this the- orem and appropriate known results about continuous embeddings, explicit characterizations of mutual almost-compact embeddings of all subtypes of Lorentz spaces are obtained. 1

Έστω C(Y,Z) το σύνολο των συνεχών συναρτήσεων από έναν τοπολογικό χώρο Υ σ' έναν τοπολογικό χώρο Ζ. Στη διπλωματική εργασία δίνουμε και μελετάμε τοπολογίες στο C(Y,Z). / Let C(Y,Z) be the set of all continuous maps from a topological space Y to a topological space Z . We give and study topologies on the set C(Y,Z) .

The first part of the thesis is devoted to the analysis on a capacity space, with capacities as substitutes of measures in the study of function spaces. The goal is to extend to the associated function lattices some aspects of the theory of Banach function spaces, to show how the general theory can be applied to classical function spaces such as Lorentz spaces, and to complete the real interpolation theory for these spaces included in [CeClM] and [Ce]. In the second part of the thesis, we present an integral inequality connecting a function space norm of the gradient of a function to an integral of the corresponding capacity of the conductor between two level surfaces of the function, which extends the estimates obtained by V. Maz’ya and S. Costea, and sharp capacitary inequalities due to V. Maz’ya in the case of the Sobolev norm. The inequality, obtained under appropriate convexity conditions on the function space, gives a characterization of Sobolev type inequalities involving two measures, necessary and sufficient conditions for Sobolev isocapacitary type inequalities, and self-improvements for integrability of Lipschitz functions. / La primera part està dedicada a l’anàlisi d’un espai de capacitat, amb capacitats com a substituts de les mesures en l’estudi d’espais de funcions. L’objectiu és estendre als recicles de funcions associats alguns aspectes de la la teoria d’espais de funcions de Banach, mostrar com la teoria general pot ser aplicada a espais funcionals clàssics com els espais de Lorentz, i completar la teoria d’interpolació real d’aquests espais inclosos en [CeClM] i [Ce]. A la segona part de la tesi es presenta una desigualtat integral que connecta la norma del gradient d’una funció en un espai de funcions amb la integral de la corresponent capacitat del conductor entre dues superfícies de nivell de la funció, que estén les estimacions obtingudes per V. Maz’ya i S. Costea, i desigualtats capacitàries fortes de V. Maz’ya en el cas de la norma de Sobolev. La desigualtat, obtinguda sota condicions de convexitat pel espai funcional, permet una caracterització de les desigualtats de tipus Sobolev per dues mesures, condicions necessàries i suficients per desigualtats isocapacitàries de tipus Sobolev, i la millora de l’autointegrabilitat de les funcions de Lipschitz.