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Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. (English) Zbl 0932.42021
Using orthogonal polynomials (i.e., ultraspherical polynomials), the authors construct families of ${C}^{0}$ and ${C}^{1}$ orthogonal, compactly supported spline multiwavelets of ${L}^{2}\left(ℝ\right)$ with various approximation orders. In the case of symmetric or antisymmetric multiscaling functions, this method allows the construction of symmetric or antisymmetric multiwavelets. By restriction on $\left[0,1\right]$, these results yield ${C}^{0}$ and ${C}^{1}$ spline multiwavelet bases of ${L}^{2}\left[0,1\right]$. A ${C}^{2}$ compactly supported spline multiwavelet basis of ${L}^{2}\left(ℝ\right)$ is also sketched, but the formulas become very complicated.
##### MSC:
 42C40 Wavelets and other special systems 41A15 Spline approximation 33C55 Spherical harmonics