\(C^1\) spline wavelets on triangulations. (English) Zbl 1126.41006
The theme of this paper is the construction of nested spline spaces based on quadratic splines and on triangulations. For this, nested families of spline spaces are built by using the famous Powell-Sabin split which is suitable for nestedness, unlike other partitioning methods. The splines are made to be continuously differentiable and the nested spaces form a multiresolution analysis. With such a multiresolution analysis, wavelets may be constructed and accordingly a general theory of wavelets on these spaces is given. The function spaces which are most useful for this type of construction are Besov spaces and indeed a whole introductory chapter is devoted to the Besov spaces on triangulations. The mentioned wavelets are constructed and shown to be stable as well, which is, of course, an important property for the usefulness of the wavelets.
Reviewer: Martin D. Buhmann (Gießen)
MSC:
| 41A15 | Spline approximation |
| 41A63 | Multidimensional problems |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 65D07 | Numerical computation using splines |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
splines; wavelets; \(C^1\) spline wavelets; general triangulations; three-direction meshes; Riesz bases; Sobolev spaces; Besov spacesReferences:
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