scholarly journals Construction of non separable dyadic compactly supported orthonormal wavelet bases for $L^2 (\mathbb R^2)$ of arbitrarily high regularity

1999 ◽  
pp. 37-58 ◽  
Author(s):  
Antoine Ayache
Keyword(s):  
Orthonormal Wavelet ◽  
Wavelet Bases ◽  
Compactly Supported ◽  
High Regularity

A new family of orthonormal wavelet bases

1998 ◽  
Vol 72 (5) ◽  
pp. 294-303 ◽  
Author(s):  
L. T. Liu ◽  
H. T. Hsu ◽  
B. X. Gao
Keyword(s):  
Orthonormal Wavelet ◽  
Wavelet Bases ◽  
New Family

scholarly journals Orhonormal Wavelet Bases on The 3D Ball Via Volume Preserving Map from the Regular Octahedron

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 994 ◽  
Author(s):  
Adrian Holhoş ◽  
Daniela Roşca
Keyword(s):  
Unit Ball ◽  
Multiresolution Analysis ◽  
Uniform Grid ◽  
Orthonormal Wavelet ◽  
Wavelet Bases ◽  
Piecewise Constant ◽  
Regular Octahedron ◽  
Piecewise Constant Functions ◽  
Local Support ◽  
Volume Preserving

We construct a new volume preserving map from the unit ball B 3 to the regular 3D octahedron, both centered at the origin, and its inverse. This map will help us to construct refinable grids of the 3D ball, consisting in diameter bounded cells having the same volume. On this 3D uniform grid, we construct a multiresolution analysis and orthonormal wavelet bases of L 2 ( B 3 ) , consisting in piecewise constant functions with small local support.

STABILITY OF BIORTHOGONAL WAVELET BASES

2003 ◽  
Vol 01 (01) ◽  
pp. 75-92
Author(s):  
PAUL F. CURRAN ◽  
GARY McDARBY
Keyword(s):  
Linear Operator ◽  
Lifting Scheme ◽  
Simple Eigenvalue ◽  
Single Parameter ◽  
Multiple Eigenvalue ◽  
Scaling Functions ◽  
Biorthogonal Wavelet ◽  
Wavelet Bases ◽  
Compactly Supported

We investigate the lifting scheme as a method for constructing compactly supported biorthogonal scaling functions and wavelets. A well-known issue arising with the use of this scheme is that the resulting functions are only formally biorthogonal. It is not guaranteed that the new wavelet bases actually exist in an acceptable sense. To verify that these bases do exist one must test an associated linear operator to ensure that it has a simple eigenvalue at one and that all its remaining eigenvalues have modulus less than one, a task which becomes numerically intensive if undertaken repeatedly. We simplify this verification procedure in two ways: (i) we show that one need only test an identifiable half of the eigenvalues of the operator, (ii) we show that when the operator depends upon a single parameter, the test first fails for values of that parameter at which the eigenvalue at one becomes a multiple eigenvalue. We propose that this new verification procedure comprises a first step towards determining simple conditions, supplementary to the lifting scheme, to ensure existence of the new wavelets generated by the scheme and develop an algorithm to this effect.

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