Standard pairs and existence of symmetric multiscaling functions
| Autor: | M. C. Lineesh, A. T. Mithun |
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| Rok vydání: | 2019 |
| Předmět: |
Basis (linear algebra)
Applied Mathematics 020206 networking & telecommunications 02 engineering and technology Function (mathematics) Physics::Data Analysis Statistics and Probability Condensed Matter::Disordered Systems and Neural Networks Matrix polynomial Nonlinear Sciences::Chaotic Dynamics Signal Processing Condensed Matter::Statistical Mechanics 0202 electrical engineering electronic engineering information engineering Applied mathematics 020201 artificial intelligence & image processing Information Systems Mathematics |
| Zdroj: | International Journal of Wavelets, Multiresolution and Information Processing. 18:1950057 |
| ISSN: | 1793-690X 0219-6913 |
| DOI: | 10.1142/s0219691319500577 |
| Popis: | Construction of multiwavelets begins with finding a solution to the multiscaling equation. The solution is known as multiscaling function. Then, a multiwavelet basis is constructed from the multiscaling function. Symmetric multiscaling functions make the wavelet basis symmetric. The existence and properties of the multiscaling function depend on the symbol function. Symbol functions are trigonometric matrix polynomials. A trigonometric matrix polynomial can be constructed from a pair of matrices known as the standard pair. The square matrix in the pair and the matrix polynomial have the same spectrum. Our objective is to find necessary and sufficient conditions on standard pairs for the existence of compactly supported, symmetric multiscaling functions. First, necessary as well as sufficient conditions on the standard pairs for the existence of symbol functions corresponding to compactly supported multiscaling functions are found. Then, the necessary and sufficient conditions on the class of standard pairs, which make the multiscaling function symmetric, are derived. A method to construct symbol function corresponding to a compactly supported, symmetric multiscaling function from an appropriate standard pair is developed. |
| Databáze: | OpenAIRE |
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