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In Chapter 1, we introduce three varieties of reproducing systems—Bessel systems, frames, and Riesz bases—within the Hilbert space context and prove a number of elementary results, including qualitative characterizations of each and several results regarding the combination and partitioning of reproducing systems. In Chapter 2, we characterize when the integer lattice translations of a countable collection of square integrable functions forms a Bessel system, a frame, and a Riesz basis. In Chapter 3, we introduce composite wavelet systems and generalize several well-known classical wavelet system results—including those regarding pointwise values of the Fourier transform of the wavelet and scaling function and those regarding dependencies on the multiresolution analysis defining properties—to the composite case. Two corollaries of these results are the nonexistence of composite scaling multifunctions of Haar-type, when the composite dilation group is infinite, and the nonexistence of classical multiwavelets, when the dilation matrix is integral and has determinant 1 in

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The Nonexistence of Shearlet-Like Scaling Multifunctions that Satisfy Certain Minimally Desirable Properties and Characterizations of the Reproducing Properties of the Integer Lattice Translations of