A Systematic Construction of Multiwavelets on the Unit Interval
Michelle Michelle · ERA: Education and Research Archive (University of Alberta) · 2017
One main goal of this thesis is to bring forth a systematic and simple construction of a multiwavelet basis on a bounded interval. The construction that we present possesses orthogonality in the derivatives of the multiwavelet basis among all scale levels. Since we are mainly interested in Riesz wavelets, we call such wavelets mth derivative--orthogonal Riesz wavelets. Furthermore, we present some necessary and sufficient conditions as to when such a construction can be done. We show that our constructed multiwavelet bases possess many desirable properties such as symmetry, stability, and short support. The second goal of this thesis is to provide some conditions that guarantee a Riesz wavelet in L_{2}(R) can be adapted so that it forms a Riesz wavelet for L_{2}(I), where I is a bounded interval. As the third goal of this thesis, we also evaluate the performance of the newly constructed bases in obtaining the numerical solutions to some differential equations to showcase their potential usefulness. More specifically, we show how the resulting coefficient matrices are sparse and have a low condition number.