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Ten lectures on wavelets
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SIAM Journal on Numerical Analysis
Hierarchical conforming finite element methods for the biharmonic equation
SIAM Journal on Numerical Analysis
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Journal of Computational and Applied Mathematics
Multiresolution analysis for surfaces of arbitrary topological type
Multiresolution analysis for surfaces of arbitrary topological type
Multiresolution analysis for surfaces of arbitrary topological type
ACM Transactions on Graphics (TOG)
SIAM Journal on Mathematical Analysis
From wavelets to multiwavelets
Proceedings of the international conference on Mathematical methods for curves and surfaces II Lillehammer, 1997
The lifting scheme: a construction of second generation wavelets
SIAM Journal on Mathematical Analysis
Multivariate refinement equations and convergence of subdivision schemes
SIAM Journal on Mathematical Analysis
Stabilizing the Hierarchical Basis by Approximate Wavelets II: Implementation and Numerical Results
SIAM Journal on Scientific Computing
Subdivision of uniform Powell—Sabin splines
Computer Aided Geometric Design
Piecewise Quadratic Approximations on Triangles
ACM Transactions on Mathematical Software (TOMS)
The Sobolev regularity of refinable functions
Journal of Approximation Theory
Approximation properties and construction of Hermite interpolants and biorthogonal mutliwavelets
Journal of Approximation Theory
Computing the Smoothness Exponent of a Symmetric Multivariate Refinable Function
SIAM Journal on Matrix Analysis and Applications
A Stabilized Lifting Construction of Wavelets on Irregular Meshes on the Interval
SIAM Journal on Scientific Computing
Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets
SIAM Journal on Matrix Analysis and Applications
Triangular √3-subdivision schemes: the regular case
Journal of Computational and Applied Mathematics
Design of Hermite Subdivision Schemes Aided by Spectral Radius Optimization
SIAM Journal on Scientific Computing
ACM Transactions on Graphics (TOG)
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Wavelets have been used in a broad range of applications such as image processing, computer graphics and numerical analysis. The lifting scheme provides an easy way to construct wavelet bases on meshes of arbitrary topological type. In this paper we shall investigate the Riesz stability of compactly supported (multi-) wavelet bases that are constructed with the lifting scheme on regularly refined meshes of arbitrary topological type. More particularly we are interested in the Riesz stability of a standard two-step lifted wavelet transform consisting of one prediction step and one update step. The design of the update step is based on stability considerations and can be described as local semiorthogonalization, which is the approach of Lounsbery et al. in their groundbreaking paper [M. Lounsbery, T.D. DeRose, J. Warren, Multiresolution analysis for surfaces of arbitrary topological type, ACM Trans. Graphics 16 (1) (1997) 34-73]. Riesz stability is important for several wavelet based numerical algorithms such as compression or Galerkin discretization of variational elliptic problems. In order to compute the exact range of Sobolev exponents for which the wavelets form a Riesz basis one needs to determine the smoothness of the dual system. It might occur that the duals, that are only defined through a refinement relation, do not exist in L"2. By using Fourier techniques we can estimate the range of Sobolev exponents for which the wavelet basis forms a Riesz basis without explicitly using the dual functions. Several examples in one and two dimensions are presented. These examples show that the lifted wavelets are a Riesz basis for a larger range of Sobolev exponents than the corresponding non-updated hierarchical bases but, in general, they do not form a Riesz basis of L"2.