| Summary: | The research in this PhD thesis is focused on some problems of general interest in both applied mathematics and engineering sciences. It contains new results, which can be directly used for solving some important structural problems in engineering sciences, but which are also of interest in pure mathematics. The main body of this PhD thesis consists of six papers (Papers A – F). In Paper A we present and discuss some recent developments concerning operational modal analysis (OMA) techniques, and also give a concrete example where the most popular OMA techniques have been implemented and applied on a steel truss bridge located over the Åby river in Sweden. In Paper B a basic mathematical model of vibrating structure is presented. We also review and compare some signal processing techniques that are of great importance for OMA and structural health monitoring (SHM) of civil engineering structures. A new application of OMA on a high rise building in Sweden, where some of the most popular OMA techniques are applied is included in this paper. In Paper C we prove and discuss some new Fourier inequalities in the generalized Lorentz type spaces, and in the important case with unbounded orthogonal systems. The derived results generalize, complement and unify several results in the literature for this general case. In Paper D we further compliment and develop the results in Paper C. In this paper we also prove and discuss the corresponding Jackson-Nikol’skii type inequalities, still in the important case with unbounded orthonormal systems. In Paper E we discuss some signal processing problems in a Bayesian framework and semi-group theory in the general case with non-separable function spaces. In particular, this is done for the case of an abstract Cauchy problem, with initial data in a non-separable Morrey space. In Paper F we present a brief description of the Hålogaland suspension bridge, along with some challenges which have already appeared. Moreover, the problems and challenges which have appeared in a number of bridges of this type in the past are also reported on and discussed. The aim of this Paper is that it can serve as a basis for our planned future research concerning SHM of this bridge. These new results are put into a more general frame in an introduction, where, in particular, some important information and challenges connected to the Hålogaland bridge in Narvik are discussed in the light of this frame.
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