M-CHANNEL MRA AND APPLICATION TO ANISOTROPIC SOBOLEV SPACES

2005 ◽  
Vol 03 (02) ◽  
pp. 153-166
Author(s):  
ELENA CORDERO
Keyword(s):  
Tensor Product ◽  
Sobolev Spaces ◽  
Unit Interval ◽  
Biorthogonal Wavelet ◽  
Bernstein Type ◽  
Wavelet Bases ◽  
Compactly Supported ◽  
Anisotropic Sobolev Spaces ◽  
Dilation Factor ◽  
Bernstein Type Inequalities

In this paper we construct compactly supported biorthogonal wavelet bases of the interval, with dilation factor M. Next, the natural MRA on the cube arising from the tensor product of a multilevel decomposition of the unit interval is developed. New Jackson and Bernstein type inequalities are proved, providing a characterization for anisotropic Sobolev spaces.

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.

scholarly journals Compactly supported wavelet bases for Sobolev spaces

2003 ◽  
Vol 15 (3) ◽  
pp. 224-241 ◽  
Author(s):  
Rong-Qing Jia ◽  
Jianzhong Wang ◽  
Ding-Xuan Zhou
Keyword(s):  
Sobolev Spaces ◽  
Wavelet Bases ◽  
Compactly Supported

Riesz basis of wavelets constructed from trigonometric B-splines

2015 ◽  
Vol 13 (04) ◽  
pp. 419-436
Author(s):  
Mahendra Kumar Jena
Keyword(s):  
Differential Operator ◽  
Sobolev Space ◽  
Sobolev Spaces ◽  
Riesz Basis ◽  
Scaling Function ◽  
Wavelet Bases ◽  
B Splines ◽  
Compactly Supported ◽  
The Scaling Function

In this paper, we construct a class of compactly supported wavelets by taking trigonometric B-splines as the scaling function. The duals of these wavelets are also constructed. With the help of these duals, we show that the collection of dilations and translations of such a wavelet forms a Riesz basis of 𝕃2(ℝ). Moreover, when a particular differential operator is applied to the wavelet, it also generates a Riesz basis for a particular generalized Sobolev space. Most of the proofs are based on three assumptions which are mild generalizations of three important lemmas of Jia et al. [Compactly supported wavelet bases for Sobolev spaces, Appl. Comput. Harmon. Anal. 15 (2003) 224–241].

scholarly journals Analysis of Compactly Supported Nonstationary Biorthogonal Wavelet Systems Based on Exponential B-Splines

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Yeon Ju Lee ◽  
Jungho Yoon
Keyword(s):  
Riesz Bases ◽  
Refinable Functions ◽  
Biorthogonal Wavelet ◽  
Wavelet Bases ◽  
Refinable Function ◽  
B Splines ◽  
Compactly Supported ◽  
Mathematical Properties ◽  
The Stability

This paper is concerned with analyzing the mathematical properties, such as the regularity and stability of nonstationary biorthogonal wavelet systems based on exponential B-splines. We first discuss the biorthogonality condition of the nonstationary refinable functions, and then we show that the refinable functions based on exponential B-splines have the same regularities as the ones based on the polynomial B-splines of the corresponding orders. In the context of nonstationary wavelets, the stability of wavelet bases is not implied by the stability of a refinable function. For this reason, we prove that the suggested nonstationary wavelets form Riesz bases for the space that they generate.

scholarly journals Biorthogonal multiresolution analyses and decompositions of Sobolev spaces

2001 ◽  
Vol 28 (9) ◽  
pp. 517-534
Author(s):  
Abdellatif Jouini ◽  
Khalifa Trimèche
Keyword(s):  
Sobolev Spaces ◽  
Biorthogonal Wavelet ◽  
Wavelet Bases

The object of this paper is to construct extension operators in the Sobolev spacesHk(]−∞,0])andHk([0,+∞[)(k≥0). Then we use these extensions to get biorthogonal wavelet bases inHk(ℝ). We also give a construction inL2([−1,1])to see how to obtain boundaries functions.

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