Abstract
We define wavelets and the wavelet transform. After discussing their basic properties, we focus on orthonormal bases of wavelets, in particular bases of wavelets with finite support.
‘Bevoegdverklaard Navorser’ at the Belgium National Foundation for Scientific Research (on leave); on leave also from Department of Theoretical Physics, Vrije Universiteit Brussel, (Belgium).
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References
See e.g. J. Morlet, G. Arens, I. Fourgeau and D. Giard, “Wave propagation and sampling theory,” Geophysics 47 (1982) 203–236.
A. Grossman and J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape”, SIAM J. Math. Anal. 15 (1984) 723–736.
P. Goupillaud, A. Grossmann and J. Morlet, “Cycle-octave and related transforms in seismic signal analysis”, Geoexploration 23 (1984) 85.
The wavelet transform is implicitly used in A. Calderón, “Intermediate spaces and interpolation, the complex method”, Studia Math. 24 (1964) 113–190.
It appears more explicitly in e.g. A. Calderón and A. Torchinsky, “Parabolic maximal functions associated to a distribution, I”, Adv. Math. 16 (1975) 1–64.
An application to a singular integral operator relevant for quantum mechanics can be found in C. Fefferman and R. de la Llave, “Relativistic stability of matter,” Rev. Mat. Iberoamericana 2 (1986).
J. R. Klauder and B.-S. Skagerstam, “Coherent States”, World Scientific (Singapore) 1985.
E. W. Aslaksen and J. R. Klauder, “Unitary representations of the affine group,” J. Math. Phys. 9 (1968) 206–211
“Continuous representation theory using the affine group”, J. Math. Phys. 10 (1969) 2267–2275.
T. Paul, “Affine coherent states and the radial Schrodinger equation I. Radial harmonic oscillator and the hydrogen atom”, to be published.
K. G. Wilson and J. B. Kogut, Physics Reports 12C (1974) 77.
J. Glimm and A. Jaffe, “Quantum physics: a functional integral point of view”. Springer (New York) 1981.
G. Battle and P. Federbush, “Ondelettes and phase cell cluster expansions: a vindication”. Comm. Math. Phys. 109 (1987) 417–419.
I. Daubechies, “The wavelet transform, time-frequency localisation and signal analysis”, to be published in IEEE Trans. Inf. Theory.
R. Balian, “Un principe d’incertitude fort en théorie du signal ou en mécanique quantique”, C. R. Acad. Se. Paris 292, série 2 (1981) 1357–1362.
F. Low, “Complex sets of wave-packets” in “A passion for physics — Essays in honor of G. Chew”, World Scientific (Singapore) 1985, pp. 17–22.
G. Battle, “Heisenberg proof of the Balian-Low theorem”, to be published in Lett. Math.Phys.
D. Gabor, “Theory of communication”, J. Inst. Elec. Eng. (London) 93 III (1946) 429–457.
M. J. Bastiaans, “A sampling theorem for the complex spectrogram and Gabor’s expansion of a signal in Gaussian elementary signals”, Optical Eng. 20 (1981) 594–598.
A.J.E. M. Janssen, “Gabor representation of generalized functions,” J. Math. Appl. 80 (1981) 377–394.
Y. Meyer, “Principe d’incertitde, bases hilbertiennes et algèbres d’opérateurs”. Séminaire Bourbaki, 1985–1986, nr.662.
P. G. Lemarié and Y. Meyer, “Ondelettes et bases hilbertiennes”, Rev. Mat. Iberoamericana 2 (1986) 1–18.
G. Battle, “A block spin construction of ondelettes. I: Lemarié functions,” Comm. Math. Phys. 110 (1987) 601–615.
P. Lemarié, “Ondelettes à localisation exponentielle”, to be published in J. de Math. Pures et Appl.
S. Mallat, “Multiresolution approximation and wavelets”, to be published.
S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation”, to be published in IEEE Trans, on Pattern Analysis and Machine Intelligence.
P. Burt and E. Adelson, “The Laplacian pyramid as a compact image code”, IEEE Trans. Comm. 31 (1983) 582–540
“A multiresolution spline with application to image mosaics”, ACM Trans, on Graphics, 2 (1983) 217–236.
I. Daubechies, “Orthonormal bases of compactly supported wavelets”. Comm. Pure & Appl. Math. 49 (1988) 909–996.
Y. Meyer, “Ondelettes et fonctions splines”. Séminaire E.D.P., Ecole Polytechnique, Paris, France, December 86.
M. J. Smith and D. P. Barnwell, “Exact reconstruction techniques for tree-structured subband coders”, IEEE Trans, on ASSP 34 (1986) 434–441.
D. Esteban and C. Galand, “Application of quadrature minor filters to split band voice coding schemes”, Proc. Int. Conf. ASSP (1977) 191–195.
G. Polya and G. Szego, “Aufgaben und Lehrsätse aus der Analysis” Vol. II, Springer (Berlin) 1971.
I. Daubechies and J. Lagarias, “Two-scale difference equations: I. Global regularity of solutions”, preprint AT&T Bell Laboratories.
— Two-scale difference equations: II Infinite products of matrices, local regularity and fractals.”, preprint AT&T Bell Laboratories.
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Daubechies, I. (1989). Orthonormal Bases of Wavelets with Finite Support — Connection with Discrete Filters. In: Combes, JM., Grossmann, A., Tchamitchian, P. (eds) Wavelets. Inverse Problems and Theoretical Imaging. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-97177-8_3
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DOI: https://doi.org/10.1007/978-3-642-97177-8_3
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