Macro-element Hierarchical Riesz Bases

  • Oleg Davydov
  • Wee Ping Yeo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8177)


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 (Ω), \(1<s<r+\frac{3}{2}\), and \(H^s_0(\Omega)\), \(1<s<\sigma+\frac{3}{2}\), \(s\notin\mathbb{Z}+\frac{1}{2}\), 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 σ. In addition, we provide a brief review of the existing constructions of C 1 Largange type hierarchical bases.


Hierarchical bases Riesz bases macro-elements bivariate splines Jackson inequality Bernstein inequality 


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Oleg Davydov
    • 1
  • Wee Ping Yeo
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of StrathclydeGlasgowScotland

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