International Conference on Multivariate Approximation
September 24-27, 2011

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Hierarchical Riesz bases for Sobolev spaces on polygonal domains

Oleg Davydov
We present a new construction of Riesz bases for the spaces $H^s(\Omega)$, $1 < s < 5/2$, relying on Lagrange interpolation by $C^1$ piecewise quadratic polynomials. In contrast to earlier results in [1], an arbitrary triangulation of the polygonal domain $\Omega$ can be used, which facilitates the numerical implementation. Similar to [2], mixed Powell-Sabin 6- and 12-splits are employed to generate suitable spline spaces, but the stability range of our wavelets is larger. The results are obtained jointly with Wee Ping Yeo. \medskip {\small [1] O.Davydov and R.Stevenson, Hierarchical Riesz Bases for $H^s(\Omega)$, $1 < s < 5/2$, Constr. Approx. 43 (2005), 365-394. [2] R.-Q.Jia, S.-T.Liu: $C^1$ spline wavelets on triangulations. Math. Comput. 77 (2008), 287-312. }