Wavelet theory and some of its applications

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 other too, 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.