In Chapters 1 and 2, we introduce the definition of interpolating refinable function vectors in dimension one and high dimensions, characterize such interpolating refinable function vectors in terms of their masks, and derive their sum rule structure explicitly. We study biorthogonal refinable function vectors from interpolating refinable function vectors. We also study the symmetry property of an interpolating refinable function vector and characterize a symmetric interpolating refinable function vector in any dimension with respect to certain symmetry group in terms of its mask. Examples of interpolating refinable function vectors with some desirable properties, such as orthogonality, symmetry, compact support, and so on, are constructed according to our characterization results. In Chapters 3 and 4, we turn to the study of general matrix extension problems with symmetry for the construction of orthogonal and biorthogonal multiwavelets. We give characterization theorems and develop step-by-step algorithms for matrix extension with symmetry. To illustrate our results, we apply our algorithms to several examples of interpolating refinable function vectors with orthogonality or biorthogonality obtained in Chapter 1. In Chapter 5, we discuss some possible future research topics on the subjects of matrix extension with symmetry in high dimensions and frequency-based non-stationary tight wavelet frames with directionality. We demonstrate that one can construct a frequency-based tight wavelet frame with symmetry and show that directional analysis can be easily achieved under the framework of tight wavelet frames. Potential applications and research directions of such tight wavelet frames with directionality are discussed. / Applied Mathematics

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