Non-stationary Sibling Wavelet Frames on Bounded Intervals
Frame Theory is a modern branch of Harmonic Analysis. It has its roots in Communication Theory and Quantum Mechanics. Frames are overcomplete and stable families of functions which provide non-unique and non-orthogonal series representations for each element of the space. The first milestone was set 1946 by Gabor with the paper ''Theory of communications''. He formulated a fundamental approach to signal decomposition in terms of elementary signals generated by translations and modulations of a Gaussian. The frames for Hilbert spaces were formally defined for the first time 1952 by Duffin and Schaeffer in their fundamental paper ''A class of nonharmonic Fourier series''. The breakthrough of frames came 1986 with Daubechies, Grossmann and Meyer's paper ''Painless nonorthogonal expansions''. Since then a lot of scientists have been investigating frames from different points of view. In this thesis we study non-stationary sibling frames, in general, and the possibility to construct such function families in spline spaces, in particular. Our work follows a theoretical, constructive track. Nonetheless, as demonstrated by several papers by Daubechies and other authors, frames are very useful in various areas of Applied Mathematics, including Signal and Image Processing, Data Compression and Signal Detection. The overcompleteness of the system incorporates redundant information in the frame coefficients. In certain applications one can take advantage of these correlations. The content of this thesis can be split naturally into three parts: Chapters 1-3 introduce basic definitions, necessary notations and classical results from the General Frame Theory, from B-Spline Theory and on non-stationary tight wavelet spline frames. Chapters 4-5 describe the theory we developed for sibling frames on an abstract level. The last chapter presents our explicit construction of a certain class of non-stationary sibling spline frames with vanishing moments in L_2[a,b] which exemplifies and thus proves the applicability of our theoretical results from Chapters 4-5. As a principle of writing we did the best possible to make this thesis self-contained. Classical handbooks, recent monographs, fundamental research papers and survey articles from Wavelet, Frame and Spline Theory are cited for further - more detailed - reading. In Chapter 3 we summarize the considerations on normalized tight spline frames of Chui, He and Stoeckler and some of the results from their article ''Nonstationary tight wavelet frames. I: Bounded intervals'' (see Appl. Comp. Harm. Anal. 17 (2004), 141-197). Our work detailed in Chapters 4-6 is meant to extend and supplement their theory for bounded intervals. Chapter 4 deals with our extension of the general construction principle of non-stationary wavelet frames from the tight case to the non-tight (= sibling) case on which our present work focuses. We apply this principle in Chapter 6 in order to give a general construction scheme for certain non-stationary sibling spline frames of order m with L vanishing moments (m \in N, m >= 2, 1 <= L <= m), as well as some concrete illustrative examples. Chapter 4 presents in Subsection 4.3.3 the motivation for our detailed investigations in Chapter 5. We need some sufficient conditions on the function families defined by coefficient matrices which are as simple as possible, in order to be able to verify easily if some concrete spline families are Bessel families (and thus sibling frames) or not. In Chapter 5 we develop general strategies for proving the boundedness of certain linear operators. These will enable us to check in Chapter 6 the Bessel condition for concrete spline systems which are our candidates for sibling spline frames. Some results concerning multivariate Bessel families are also included.