Hierarchical Riesz Bases for Hs(Ω), 1 &lt; s &lt; 5/2

Davydov, Oleg; Stevenson, Rob

doi:10.1007/s00365-004-0593-2

Cited by 15 publications

(10 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…for all − s < s < s φ which follows from standard techniques as used in [10,12,28]. This equivalence implies the H s Riesz basis property for the finite set…”

Section: Furthermore ψ Is Not a Riesz Basis In H S (R D ) For Any S mentioning

confidence: 70%

Stability analysis of biorthogonal multiwavelets whose duals are not in and its application to local semiorthogonal lifting

Maes

Bultheel

2008

Applied Numerical Mathematics

View full text Add to dashboard Cite

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 .

show abstract

“…for all − s < s < s φ which follows from standard techniques as used in [10,12,28]. This equivalence implies the H s Riesz basis property for the finite set…”

Section: Furthermore ψ Is Not a Riesz Basis In H S (R D ) For Any S mentioning

confidence: 70%

Stability analysis of biorthogonal multiwavelets whose duals are not in and its application to local semiorthogonal lifting

Maes

Bultheel

2008

Applied Numerical Mathematics

View full text Add to dashboard Cite

show abstract

“…See Figure 3d, 3e for the coefficients in univariate C 1 -, C 2 -, and C 3 -conditions across the edges v 1 , v 7 , v 6 , v 7 , and see Figure 3f for the coefficients of a univariate C 3 -condition along the line segment v 4 , v 9 . We can use either ( 5) or (6) to show this. We use (6)…”

Section: Bernstein-bézier Techniquesmentioning

confidence: 99%

A Hermite interpolatory subdivision scheme for $C^2$-quintics on the Powell-Sabin 12-split

Lyche,

Muntingh

2013

Preprint

View full text Add to dashboard Cite

In order to construct a C 1 -quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell-Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme [5]. In this paper we introduce a nodal macroelement on the 12-split for the space of quintic splines that are locally C 3 and globally C 2 . For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements.

show abstract

“…In the latter a combination of 6-and 12-splits is used. For the FVS C 1 -cubic quadrangular macro element see [6,9], and for a survey of refinable multivariate spline functions see [8].…”

Section: Introductionmentioning

confidence: 99%

A Hermite interpolatory subdivision scheme for C2-quintics on the Powell–Sabin 12-split

Lyche¹,

Muntingh²

2014

Computer Aided Geometric Design

View full text Add to dashboard Cite

In order to construct a C 1-quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell-Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme [5]. In this paper we introduce a nodal macroelement on the 12-split for the space of quintic splines that are locally C 3 and globally C 2. For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements.

show abstract

Hierarchical Riesz Bases for Hs(Ω), 1 < s < 5/2

Cited by 15 publications

References 18 publications

Stability analysis of biorthogonal multiwavelets whose duals are not in and its application to local semiorthogonal lifting

Stability analysis of biorthogonal multiwavelets whose duals are not in and its application to local semiorthogonal lifting

A Hermite interpolatory subdivision scheme for $C^2$-quintics on the Powell-Sabin 12-split

A Hermite interpolatory subdivision scheme for C2-quintics on the Powell–Sabin 12-split

Contact Info

Product

Resources

About