This thesis consists of five papers (A-E), which examine the operation of infimal convolution and discuss its close connections to unilateral analysis, convex analysis, inequalities, approximation, and optimization. In particular, we attempt to provide a detailed investigation for both the convex and the non-convex case, including several examples. Paper (A) is both a survey of and a self-contained introduction to the operation of infimal convolution. In particular, we discuss the infimal value and minimizers of an infimal convolute, infimal convolution on subadditive functions, sufficient conditions for semicontinuity or continuity of an infimal convolute, "exactness," regularizing effects, continuity of the operation of infimal convolution, and approximation methods based on infimal convolution. A Young-type inequality, closely connected to the operation of infimal convolution, is studied in paper (B). The main results obtained are an equivalence theorem and a representation formula. In paper (C) we consider coercive, convex, proper, and lower sernicontinuous functions on a reflexive Banach space. For the infimal convolution of such functions we establish, in particular, different formulae. Moreover, we demonstrate the possibility of using the formulae obtained for solving special types of Hamilton-Jacobi equations. Furthermore, the operation of infimal convolution is interpreted from a physical viewpoint. Paper (D) presents properties of infimal convolution of functions that are uniformly continuous on bounded sets. In particular, we present regularization procedures by means of infimal convolution. The role of growth conditions on the functions under consideration is essential. Finally, in paper (E) we study semicontinuity, continuity, and differentiability of the infimal convolute of two convex functions. Moreover, under certain geometric conditions, the classical Moreau-Yosida approximation process is, roughly speaking, extended to the non-convex case. / Godkänd; 1994; 20070426 (ysko)

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<p>Godkänd; 1991; 20080410 (ysko)</p>

This thesis deals with applied mathematics with wavelets as a joint subject. There is an introduction and two extensive papers, of which one is already published in an international journal. The introduction presents wavelet theory including both the discrete and continuous wavelet transforms, the corresponding Fourier transforms, wavelet packets and the uncertainty principle. Moreover, it is a guide to applications. We consider applications that are strongly connected to the thesis and also other but more briefly. Also, the connection to both of the papers is included in the introduction. Paper 1 considers irregular sampling in shift-invariant spaces, such as for instance the spaces that are connected to a multiresolution analysis within wavelet theory. We set out the necessary theoretical aspects to enable reconstruction of an irregularly sampled function. Unlike most previous work in this area the method that is proposed in Paper 1 opens up for comparatively easy calculations of examples. Accordingly, we give a thorough exposition of one example of a sampling function. Paper 2 contains derivation and comparison of several different vibration analysis techniques for automatic detection of local defects in bearings. An extensive number of mathematical methods are suggested and evaluated through tests with both laboratory and industrial environment signals. Two out of the four best methods found are wavelet based, with an error rate of about 10%. Finally, there are many potentially performance improving additions included. / Godkänd; 2005; 20061108 (ysko)

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Godkänd; 1997; 20070418 (ysko)

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