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Biorthogonal wavelets and tight framelets from smoothed pseudo splines
1Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi 710072 P.R. China.
Journal of Inequalities and Applications
|August 4, 2017
Summary
This study demonstrates the linear independence of smoothed pseudo spline shifts, enabling the derivation of biorthogonal wavelets. These wavelets are crucial for constructing tight frame systems with specific approximation orders.
Area of Science:
- Applied Mathematics
- Harmonic Analysis
- Signal Processing
Background:
- Smoothed pseudo splines are introduced via convolution for divergence-free and curl-free wavelets.
- These splines extend the concept of pseudo splines.
Purpose of the Study:
- To establish the linear independence of smoothed pseudo spline shifts.
- To generalize Riesz wavelets and derive biorthogonal wavelets from smoothed pseudo splines.
- To construct tight frame systems with desired approximation orders.
Main Methods:
- Convolution method for smoothed pseudo splines.
- Demonstration of linear independence of spline shifts.
- Unitary extension principle for tight frame construction.
Main Results:
- Proved linear independence of smoothed pseudo spline shifts.
- Derived biorthogonal wavelets from smoothed pseudo splines.
- Constructed associated tight frame systems with controlled approximation orders.
Conclusions:
- Linear independence of shifts is key for biorthogonal wavelet construction.
- Smoothed pseudo splines provide a foundation for advanced wavelet systems.
- The methods yield flexible tight frames for signal processing applications.
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