O. Davydov and W. P. Yeo, Macro-element hierarchical Riesz bases, in "Mathematical Methods for Curves and Surfaces: 8th International Conference, Oslo, 2012" (M. Floater et al, Eds.), pp.112--134, LNCS 8177, Springer-Verlag, 2014. doi:10.1007/978-3-642-54382-1_7

Abstract: We show that a nested sequence of $C^r$ macro-element spline spaces on quasi-uniform triangulations gives rise to hierarchical Riesz bases of Sobolev spaces $H^s(\Omega)$, $s\in (1,r+\frac{3}{2})$, and $H^s_0(\Omega)$, $s\in(1,\sigma+\frac{3}{2})$, $s\notin\mathbb{Z}+\frac{1}{2}$, as soon as there is a nested sequence of Lagrange interpolation sets with uniformly local and bounded basis functions, and, in case of $H^s_0(\Omega)$, the nodal interpolation operators associated with the macro-element spaces are boundary conforming of order $\sigma$. In addition, we provide a brief review of the existing constructions of $C^1$ Largange type hierarchical bases.

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