Abstract
Suppose to and t 1 are N x N real valued matrices whose joint spectral radius is less than 1. We review the construction of a family \( {F_{{t_0}}} \),t 1 of fractal interpolation functions associated with these two matrices and note that compactly supported refinable functions (and thus compactly supported wavelets) are pieced together functions from \( {F_{{t_0}}} \),t 1 for some t 0, t 1. The so-called HD (Humpty-Dumpty) problem is to find all refinable (vector) functions that can be constructed from given t 0,t 1 and we review and collect recent results by the author and T. Hogan. We then use these results to provide a complete classification of continuous refinable functions with approximation order 2 and local dimension 3.
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Hardin, D.P. (2002). Wavelets are Piecewise Fractal Interpolation Functions. In: Barnsley, M.F., Saupe, D., Vrscay, E.R. (eds) Fractals in Multimedia. The IMA Volumes in Mathematics and its Application, vol 132. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9244-6_6
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DOI: https://doi.org/10.1007/978-1-4684-9244-6_6
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