Abstract
The aim of this lecture is to give a quick review on wavelets and spline theory, a very quick one since Professor Chui already gave you an extended lecture on spline wavelets [7]. To avoid too much redundancy with Chui’s talk, I will speak as few as possible about “classical wavelet theory” - namely the orthonormal wavelet bases provided by the multiresolution analysis scheme - and a little more about heretical wavelet theories, such as bi-orthogonal wavelets or the pre-wavelets of G. Battle. A very nice example of how to apply such heretical wavelets will be given in the study of divergence-free vector wavelets.
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References
A. Aldroubi, M. Eden & M. Unser A family of polynomial spline wavelet transform. Preprint, 1990.
P. Auscher Ondelettes fractales et applications. Thèse, Paris IX, 1989.
G. Battle A block spin construction of ondelettes. Part I: Lemarié functions. Comm. Math. Phys. 110 (1987), 601–615.
G. Battle A block spin construction of ondelettes. Part II: The QFT connection. Comm. Math. Phys. 114 (1988), 93–102.
G. Battle & P. Federbush A note on divergence-free vector wavelets. Preprint, T.A.M.U., 1991.
G. Battle & P. Federbush Divergence-free vector wavelets. Preprint, T.A.M. U., 1991.
C. K. Chui An introduction to spline wavelets. Lecture at the NATO-ASI on “Approximation theory, splines and applications”, Maratea, 1991.
C. K. Chui & J. Z. Wang On compactly supported spline wavelets and a duality principle. To appear in Trans. Amer. Math. Soc.
C. K. Chui & J. Z. Wang A cardinal spline approach to wavelets. To appear in Proc. Amer. Math. Soc.
A. Cohen, I. Daubechies & J. C. Feauveau Bi-orthogonal bases of compactly supported wavelets. Preprint, ATT & Bell Laboratories, 1990.
I. Daubechies Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 46 (1988), 909–996.
I. Daubechies & J. Lagarias Two-scale difference equations. Preprint, ATT & Bell Laboratories, 1989.
J. C. Feauveau Analyse multi-resolution par ondelettes non orthogonales et banc de filtres numériques. Thèse, Paris XI, 1990.
S. Jaffard Construction et propriétés des bases d’ondelettes. Thèse, Ecole Polytechnique, 1989.
P. G. Lemarie Construction d’ondelettes splines. Unpublished, 1987.
P. G. Lemarie Ondelettes à localisation exponentielle. J. Math. Pures & Appl. 67 (1988), 227–236.
P. G. Lemarie Théorie L 2 des surfaces splines. Unpublished 1987.
P. G. Lemarie Bases d’ondelettes sur les groupes de Lie stratifiés. Bull. Soc. Math. France 117 (1989), 211–232.
P. G. Lemarie Some remarks on wavelets and interpolation theory. Preprint, Paris XI, 1990.
P. G. Lemarie Fonctions à support compact dans les analyses multi-résolutions. To appear in Revista Matematica Ibero-americana.
P. G. Lemarie-Rieusset Analyses multi-résolutions non orthogonales et ondelettes vecteurs à divergence nulle. Preprint, Paris XI, 1991.
P. G. Lemarie & Y. Meyer Ondelettes et bases hilbertiennes. Rev. Mat. Iberoamericana 2 (1986), 1–18.
S. Mallat A theory for multi-resolution signal decomposition: the wavelet representation. IEEE PAMI 11 (1989), 674–693.
Y. Meyer Ondelettes et opérateurs, tome 1. Paris, Hermann, 1990.
I. J. Schoenberg Cardinal spline interpolation, CBMS-NSF. Series in Applied Math. ≠ 12, SIAM Publ., Philadelphia, 1973.
J. O. Stromberg A modified Franklin system and higher-order systems of ℝn as unconditional bases for Hardy spaces, in Conf. on Harmonic Anal. in honor of A. Zygmund, vol. 2, Waldsworth, 1983, 475–494.
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© 1992 Springer Science+Business Media Dordrecht
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Lemarie-Rieusset, PG. (1992). Wavelets, Splines and Divergence-Free Vector Functions. In: Singh, S.P. (eds) Approximation Theory, Spline Functions and Applications. NATO ASI Series, vol 356. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2634-2_25
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DOI: https://doi.org/10.1007/978-94-011-2634-2_25
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