Abstract
Because refinable functions play a central role in the study and construction of affine systems of framelets and wavelets, in this chapter we study several classes of special refinable functions. As a consequence, we provide complete analysis of framelets and wavelets (or more precisely, multiframelets and multiwavelets) that are derived from refinable (vector-valued) functions. In particular, we investigate refinable functions having analytic expressions (such as piecewise polynomials), refinable Hermite interpolants, refinable orthogonal functions, and refinable biorthogonal functions. We characterize dual framelets and biorthogonal wavelets in Sobolev spaces that are derived from refinable functions.
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