Scientific Research

An Academic Publisher

Wavelet Bases Made of Piecewise Polynomial Functions: Theory and Applications

We present wavelet bases made of piecewise (low degree) polynomial functions with an (arbitrary) assigned number of vanishing moments. We study some of the properties of these wavelet bases; in particular we consider their use in the approximation of functions and in numerical quadrature. We focus on two applications: integral kernel sparsification and digital image compression and reconstruction. In these application areas the use of these wavelet bases gives very satisfactory results.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

L. Fatone, M. Recchioni and F. Zirilli, "Wavelet Bases Made of Piecewise Polynomial Functions: Theory and Applications,"

*Applied Mathematics*, Vol. 2 No. 2, 2011, pp. 196-216. doi: 10.4236/am.2011.22022.

[1] | S. Mallat, “Multiresolution Approximation and Wavelets,” Transactions of the American Mathematical Society, Vol. 315, 1989, pp. 69-88. |

[2] | S. Mallat, “Multifrequency Channel Decompositions of Images and Wavelet Models,” IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 37, 1989, pp. 2091-2110. doi:10.1109/29.45554 |

[3] | Y. Meyer, “Ondelettes, Fonctions Splines et Analyses Graduées,” Rendiconti del Seminario Matematico Università Politecnico di Torino, Vol. 45, 1988, pp. 1-42. |

[4] | Y. Meyer, “Ondelettes et Opérateurs I: Ondelettes,” Hermann, Paris, 1990. |

[5] | Y. Meyer, “Ondelettes et Opérateurs II: Opérateurs de Calderón-Zygmund,” Hermann, Paris, 1990. |

[6] | R. R. Coifman and Y. Meyer, “Ondelettes et Opérateurs III: Opérateurs multilinéaires,” Hermann, Paris, 1991. |

[7] | I. Daubechies, “Orthonormal Bases of Compactly Supported Wavelets,” Communications on Pure and Applied Mathematics, Vol. 41, No. 7, 1998, pp. 909-996. doi:10.1002/cpa.3160410705 |

[8] | I. Daubechies, “Ten Lectures on Wavelets,” SIAM, Philadelphia, 1992. |

[9] | B. K. Alpert, “Wavelets and Other Bases for Fast Numerical Linear Algebra,” In: C. K. Chui, Ed., Wavelets: A Tutorial in Theory and Applications, Academic Press, New York, 1992, pp. 181-216. |

[10] | B. K. Alpert, “A Class of Bases in L2 for the Sparse Representation of Integral Operators,” SIAM Journal on Mathematical Analysis, Vol. 24, No. 1, 1993, pp. 246-262. doi:10.1137/0524016 |

[11] | L. Fatone, M. C. Recchioni and F. Zirilli, “New Scattering Problems and Numerical Methods in Acoustics,” In: S. G. Pandalai, Recent Research Developments in Acoustics, Transworld Research Network, Kerala, Vol. 2, 2005, pp. 39-69. |

[12] | L. Fatone, G. Rao, M. C. Recchioni and F. Zirilli, “High Performance Algorithms Based on a New Wavelet Expansion for Time Dependent Acoustic Obstacle Scattering,” Communications in Computational Physics, Vol. 2, No. 6, 2007, pp. 1139-1173. |

[13] | C. A. Micchelli and Y. Yu, “Using the Matrix Refinement Equation for the Construction of Wavelets on Invariant Sets,” Applied and Computational Harmonic Analysis, Vol. 1, No. 4, 1994, pp. 391-401. doi:10.1006/acha.1994.1024 |

[14] | C. A. Micchelli and Y. Yu, “Reconstruction and Decomposition Algorithms for Biorthogonal Multi-Wavelets,” Multidimensional Systems and Signal Processing, Vol. 8, No. 1-2, 1997, pp. 31-69. doi:10.1023/A:1008264805830 |

[15] | B. K. Alpert, G. Beylkin, R. R. Coifman and V. Rokhlin, “Wavelet-Like Bases for the Fast Solution of Second-Kind Integral Equations,” SIAM Journal on Scientific Computing, Vol. 14, No. 1, 1993, pp. 159-184. doi:10.1137/0914010 |

[16] | L. Fatone, M. C. Recchioni and F. Zirilli, “A Numerical Method for Time Dependent Acoustic Scattering Problems Involving Smart Obstacles and Incoming Waves of Small Wavelengths,” In: B. Nilsson and L. Fishman, Eds., Mathematical Modeling of Wave Phenomena, AIP Conference Proceedings, Khanty-Mansiysk, Vol. 834, 17-22 July 2006, pp. 108-121. |

[17] | W. Rudin, “Real and Complex Analysis,” McGraw Hill Inc., New York, 1966. |

[18] | J. Stoer and R. Bulirsch, “Introduction to Numerical Analysis,” Springer-Verlag, New York, 2002. |

[19] | G. Beylkin, R. R. Coifman and V. Rokhlin, “Fast Wavelet Transforms and Numerical Algorithms I,” Communications on Pure and Applied Mathematics, Vol. 44, No. 2, 1991, pp. 141-183. doi:10.1002/cpa.3160440202 |

[20] | W. Sweldens and R. Piessens, “Quadrature Formulae and Asymptotic Error Expansions for Wavelet Approximations of Smooth Functions,” SIAM Journal on Numerical Analysis, Vol. 31, No. 4, 1994, pp. 1240-1264. doi:10. 1137/0731065 |

[21] | D. Huybrechs and S. Vandewalle, “Composite Quadrature Formulae for the Approximations of Wavelet Coeffi- cients of Piecewise Smooth and Singular Functions,” Journal of Computational and Applied Mathematics, Vol. 180, No. 1, 2005, pp. 119-135. doi:10.1016/j.cam.2004.10. 005 |

[22] | W. Gautschi, L. Gori and F. Pitolli, “Gauss Quadrature Rules for Refinable Weight Functions,” Applied and Computational Harmonic Analysis, Vol. 8, No. 3, 2000, pp. 249-257. doi:10.1006/acha.1999.0306 |

[23] | A. Barinka, T. Barsch, S. Dahlke and M. Konik, “Some Remarks on Quadrature Formulas for Refinable Functions and Wavelets,” ZAMM Journal of Applied Mathematics and Mechanics, Vol. 81, No. 12, 2001, pp. 839-855. doi:10.1002/1521-4001(200112)81:12<839::AID-ZAMM839>3.0.CO;2-F |

[24] | F. Keinert, “Biorthogonal Wavelets for Fast Matrix Computations,” Applied and Computational Harmonic Analysis, Vol. 1, No. 2, 1994, pp. 147-156. doi:10.1006/acha.1994.1002 |

[25] | D. S. Taubman and M. W. Marcellin, “JPEG2000: Image Compression Fundamentals, Standards and Practice,” Kluwer Academic Publishers, Boston, 2002. |

[26] | R. A. DeVore, B. D. Jawerth and B. J. Lucier, “Image Compression through Wavelet Transform Coding,” IEEE Transactions on Information Theory, Vol. 38, No. 2, 1992, pp. 719-746. doi:10.1109/18.119733 |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.