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A novel method of constructing compactly supported orthogonal scaling functions from splines
Shouzhi Yang1, Huiqing Huang1,2
1Department of Mathematics, Shantou University, Shantou, Guangdong 515063 P.R. China.
Journal of Inequalities and Applications
|July 21, 2017
Summary
This study introduces a new method for constructing compactly supported orthogonal scaling functions using spline functions. The novel approach ensures orthogonality while preserving the compact support property of spline functions.
Area of Science:
- Applied Mathematics
- Numerical Analysis
- Signal Processing
Background:
- Orthogonal scaling functions and wavelets are crucial in signal processing and numerical analysis.
- Standard B-splines, while offering compact support, are generally not orthogonal.
- Existing methods to achieve orthogonality often sacrifice compact support.
Purpose of the Study:
- To develop a novel construction for compactly supported orthogonal scaling functions and wavelets.
- To address the limitation of non-orthogonality in standard B-splines.
- To maintain compact support while inducing orthogonality.
Main Methods:
- Utilizing the orthonormalization procedure formula.
- Employing a weighted average method.
- Constructing the two-scale symbol for the scaling functions.
Main Results:
- A new method for generating compactly supported orthogonal scaling functions is presented.
- The proposed construction successfully combines orthogonality with compact support, overcoming limitations of standard B-splines.
- The method is applicable to spline functions of various orders.
Conclusions:
- The presented construction method is effective for creating compactly supported orthogonal scaling functions.
- This work offers a valuable tool for applications requiring both properties in areas like approximation theory and data analysis.
- The findings contribute to the advancement of wavelet theory and its practical implementations.
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