| The complete characterization of orthonormal (multi)wavelets in L2( R ) is given by two equations and a norm condition. Parseval wavelets are characterized by the two equations alone, and semiorthogonal Parseval wavelets can be characterized by the two equations and a requirement of the wavelet dimension function.; The goal of this thesis is to join the study of (multi)wavelets Psi via their characteristic equations with the study of wavelets via certain collections of subspaces of L2( R ), known as generalized multiresolution analyses (GMRAs). We provide a new proof of the necessity of the characteristic equations in the case of certain semiorthogonal Parseval wavelets. GMRA theory is not unique to L2( R ), and we are able to apply our structural approach to wavelet equations in an abstract Hilbert space H . In the context of integrable GMRAs, we show that of the statements (1) Psi is a Parseval wavelet for H , (2) Psi is semiorthogonal, (3) Psi satisfies a given characteristic equation, any two imply the third. The combination of the two characteristic equations and the dimension function condition in L2( R ) is analogous to our single equation in H . We illustrate our results with four specific examples. |
