An Orthogonal Scaling Vector Generating a Space of $C^1$ Cubic Splines Using Macroelements

Bruce Kessler, Western Kentucky University

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Research was supported by the Kentucky Science and Engineering Foundation, Grant KSEF-324-RDE-003. The posted version is a preprint. The final version is published in Journal of Concrete and Applicable Mathematics: Special Issues on Wavelets and Applications, v.4 (4) (2006): 393-414.

Abstract

The main result of this paper is the creation of an orthogonal scaling vector of four differentiable functions, two supported on $[-1,1]$ and two supported on $[0,1]$, that generates a space containing the classical spline space $\s_{3}^{1}(\Z)$ of piecewise cubic polynomials on integer knots with one derivative at each knot. The author uses a macroelement approach to the construction, using differentiable fractal function elements defined on $[0,1]$ to construct the scaling vector. An application of this new basis in an image compression example is provided.