Wavelet decompositions and function spaces on the unit cube
Abstract
The main objective of this thesis is to develop a new construction of wavelet bases on the unit interval and to establish a characterization of Besov spaces on the unit cube by wavelet coefficients. This thesis consists of three parts. First, we consider several constructions of (bi-)orthogonal wavelet bases on the unit interval. From a practical point of view, these constructions have a common disadvantage. We provide a new construction of biorthogonal wavelet bases on the unit interval that avoids this common disadvantage of earlier constructions while preserving their advantages. Second, we introduce certain general families of functions that include all wavelet bases on the unit interval. These families are not necessarily derived by dilations and translations of a function. With these families, we establish Littlewood-Paley type estimates of Besov spaces. Combining these estimates and wavelet decompositions from the first part, we characterize Besov spaces on the unit cube by the wavelet coefficients. Finally, we provide a constructive wavelet decomposition of $L\sb{p}\ (1\le p\le\infty)$ into box splines. To calculate wavelet coefficients, we apply the local polynomial $L\sb2$-approximation and then quasi-interpolation techniques. Furthermore, we characterize the Besov space, $B\sbsp{q}{\alpha}(L\sb{q}),$ by the constructive wavelet coefficients with an explicit form. We also provide the characterization of the Besov space on the unit cube. DeVore et al. have independently studied this characterization with a different proof; however, they have used a nonconstructive local approximation rather than the local $L\sb2$-approximation. Our explicit formula calculating wavelet coefficients can be directly employed for numerical implementations.
Degree
Ph.D.
Advisors
Lucier, Purdue University.
Subject Area
Mathematics
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