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**Orthonormal wavelets, analysis of operators, and applications to numerical analysis.**
*(English)*
Zbl 0764.65066

Wavelets: A tutorial in theory and applications, Wavelet Anal. Appl. 2, 543-601 (1992).

[For the entire collection see Zbl 0744.00020.]

The authors give a review on algorithms for the construction of orthonormal bases of wavelets and on the main properties of the decomposition of a function with respect to these bases. This paper shows how to use these bases in order to study large classes of operators. Theoretical and numerical applications for the resolution of partial differential equations (Dirichlet boundary problems and evolution partial differential equations) are presented.

This paper is essentially self-contained. Many results are given in a simplified form, or their proofs are just sketched. The authors also mention many open problems related to the wavelet decomposition and its applications. Some numerical tests are also included.

The authors give a review on algorithms for the construction of orthonormal bases of wavelets and on the main properties of the decomposition of a function with respect to these bases. This paper shows how to use these bases in order to study large classes of operators. Theoretical and numerical applications for the resolution of partial differential equations (Dirichlet boundary problems and evolution partial differential equations) are presented.

This paper is essentially self-contained. Many results are given in a simplified form, or their proofs are just sketched. The authors also mention many open problems related to the wavelet decomposition and its applications. Some numerical tests are also included.

Reviewer: K.Najzar (Praha)

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

35K15 | Initial value problems for second-order parabolic equations |