| The construction of a multiwavelet basis in higher dimensions begins with a refinement equation of the form f(x) = sumk∈Gammac kf(Ax - k), where f : Rd→C r , Gamma is a full-rank lattice in Rn , A is a dilation matrix, and each c k is an r x r complex matrix. In one dimension, a great deal is known about how to construct wavelets with certain desirable properties, including orthogonality, symmetry, smoothness, and rapid decay.; The natural setting for many applications is in higher dimensions. These higher-dimensional constructions have proved difficult because the classical one-dimensional techniques fail dramatically in these settings. This thesis focuses on the development of new time-domain based techniques not limited by the factorization requirements or other difficulties inherent in the one-dimensional approach.; For many applications, it is desirable to construct a scaling function with a certain degree of smoothness. One tool for examining the smoothness of a refinable function is the joint spectral radius. It is shown that for a multivariate vector scaling function, if the joint spectral radius of a particular set of matrices is less than rho(A)-nu , then the scaling function is nu-times continuously differentiable.; In order to construct an orthonormal wavelet basis, the translates of the scaling function along Gamma must form an orthonormal basis for their span. In the one-dimensional, single function setting, this can be characterized by Lawton's condition, which is stated in terms of the eigenvalues of a matrix whose entries are constructed from the refinement coefficients. Conditions analogous to Lawton's condition are developed in the multivariate setting which determine if the Gamma-translates of a vector scaling function are orthonormal.; Finally, the issue of constructing the wavelets from the scaling function is addressed. If the scaling function has orthonormal translates, this is equivalent to completing an orthogonal matrix formed from the refinement coefficients. In general, it is not known if it is possible to complete this matrix, or how to do so. If D is a complete set of representatives of Gamma/ A(Gamma), it is assumed that the nonzero refinement coefficients occur in D ∪ (D + Ai) for some nonzero i ∈ Gamma. Under this assumption, a constructive algorithm is given for completing the orthogonal matrix. |
