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.
Disciplines
Applied Mathematics
Recommended Repository Citation
Kessler, Bruce. (2004). An Orthogonal Scaling Vector Generating a Space of $C^1$ Cubic Splines Using Macroelements. Journal of Concrete and Applicable Mathematics: Special Issues on Wavelets and Applications, 4 (4), 393-414.
Available at:
https://digitalcommons.wku.edu/math_fac_pub/8
Comments
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.