Shannon wavelets theory.

*(English)*Zbl 1162.42314Summary: Shannon wavelets are studied together with their differential properties (known as connection coefficients). It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction of \(L_{2}(\mathbb R)\) functions. The differential properties of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon wavelets are \(C^{\infty }\)-functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of the \(C^{\ell }\)-functions.

##### MSC:

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

##### References:

[1] | I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1992. · Zbl 0776.42018 |

[2] | C. Cattani, “Harmonic wavelets towards the solution of nonlinear PDE,” Computers & Mathematics with Applications, vol. 50, no. 8-9, pp. 1191-1210, 2005. · Zbl 1118.65133 |

[3] | C. Cattani and J. Rushchitsky, Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure, vol. 74 of Series on Advances in Mathematics for Applied Sciences, World Scientific, Hackensack, NJ, USA, 2007. · Zbl 1152.74001 |

[4] | G. Toma, “Practical test-functions generated by computer algorithms,” in Proceedings of the International Conference on Computational Science and Its Applications (ICCSA ’05), vol. 3482 of Lecture Notes in Computer Science, pp. 576-584, Singapore, May 2005. · Zbl 05377784 |

[5] | M. Unser, “Sampling-50 years after Shannon,” Proceedings of the IEEE, vol. 88, no. 4, pp. 569-587, 2000. |

[6] | A. Latto, H. L. Resnikoff, and E. Tenenbaum, “The evaluation of connection coefficients of compactly supported wavelets,” in Proceedings of the French-USA Workshop on Wavelets and Turbulence, Y. Maday, Ed., pp. 76-89, Springer, Princeton, NY, USA, June 1991. |

[7] | E. B. Lin and X. Zhou, “Connection coefficients on an interval and wavelet solutions of Burgers equation,” Journal of Computational and Applied Mathematics, vol. 135, no. 1, pp. 63-78, 2001. · Zbl 0990.65096 |

[8] | J. M. Restrepo and G. K. Leaf, “Wavelet-Galerkin discretization of hyperbolic equations,” Journal of Computational Physics, vol. 122, no. 1, pp. 118-128, 1995. · Zbl 0838.65096 |

[9] | C. H. Romine and B. W. Peyton, “Computing connection coefficients of compactly supported wavelets on bounded intervals,” Tech. Rep. ORNL/TM-13413, Computer Science and Mathematical Division, Mathematical Sciences Section, Oak Ridge National Laboratory, Oak Ridge, Tenn, USA, http://www.ornl.gov/ webworks/cpr/rpt/91836.pdf. |

[10] | C. Cattani, “Harmonic wavelet solutions of the Schrödinger equation,” International Journal of Fluid Mechanics Research, vol. 30, no. 5, pp. 1-10, 2003. |

[11] | C. Cattani, “Connection coefficients of Shannon wavelets,” Mathematical Modelling and Analysis, vol. 11, no. 2, pp. 117-132, 2006. · Zbl 1117.65179 |

[12] | C. Cattani, “Shannon wavelet analysis,” in Proceedings of the 7th International Conference on Computational Science (ICCS ’07), Y. Shi, G. D. van Albada, J. Dongarra, and P. M. A. Sloot, Eds., vol. 4488 of Lecture Notes in Computer Science, pp. 982-989, Springer, Beijing, China, May 2007. · Zbl 05293700 |

[13] | S. V. Muniandy and I. M. Moroz, “Galerkin modelling of the Burgers equation using harmonic wavelets,” Physics Letters A, vol. 235, no. 4, pp. 352-356, 1997. · Zbl 1044.65511 |

[14] | D. E. Newland, “Harmonic wavelet analysis,” Proceedings of the Royal Society of London A, vol. 443, no. 1917, pp. 203-225, 1993. · Zbl 0793.42020 |

[15] | W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation, and Statistical Applications, vol. 129 of Lecture Notes in Statistics, Springer, New York, NY, USA, 1998. · Zbl 0899.62002 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.