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Construction of Wavelets via MRA

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Lecture Notes on Wavelet Transforms

Part of the book series: Compact Textbooks in Mathematics ((CTM))

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Abstract

The idea of multiresolution analysis (MRA) was proposed by Stéphane Mallat and Yves Meyer in 1986, and this can be considered as a rebirth of wavelet theory. This is a new and remarkable idea which deals with a general formalism for construction of an orthogonal basis of wavelets. Indeed, MRA is central to all constructions of wavelet bases. Mallat’s brilliant work (Mallat 1989a,b,c) has been the major source of many new developments in wavelet analysis and its wide variety of applications.

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Notes

  1. 1.

    The following list includes books and research papers that have been useful for the preparation of these notes as well as some which may be of interest for further study.

Bibliography

  • Mallat, S. (1989a). Multiresolution approximations and wavelet orthonormal basis of \(L^{2}(\mathbb{R})\). Transactions of the American Mathematical Society, 315, 69–88.

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  • Mallat, S. (1989b). A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 678–693.

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  • Mallat, S. (1989c). Multi-frequency channel decompositions of images and wavelet models. IEEE Transactions on Acoustics, Speech, and Signal Processing, 37, 2091–2110.

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  • Strang, G. (1989). Wavelets and dilation equations. SIAM Review, 31, 614–627.

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, University of Texas – Rio Grande Valley, Edinburg, Texas, USA

    Lokenath Debnath

  2. Department of Mathematics, University of Kashmir, Anantnag, Jammu and Kashmir, India

    Firdous A. Shah

Authors
  1. Lokenath Debnath
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  2. Firdous A. Shah
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Debnath, L., Shah, F.A. (2017). Construction of Wavelets via MRA. In: Lecture Notes on Wavelet Transforms. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59433-0_4

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