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Construction of orthonormal wavelets using Kampé de Fériet functions

Author: Ahmed I. Zayed
Journal: Proc. Amer. Math. Soc. 130 (2002), 2893-2904
MSC (2000): Primary 42C40, 33C20; Secondary 42C15, 33E20
Published electronically: May 1, 2002
MathSciNet review: 1908912
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Abstract: One of the main aims of this paper is to bridge the gap between two branches of mathematics, special functions and wavelets. This is done by showing how special functions can be used to construct orthonormal wavelet bases in a multiresolution analysis setting. The construction uses hypergeometric functions of one and two variables and a generalization of the latter, known as Kampé de Fériet functions. The mother wavelets constructed by this process are entire functions given by rapidly converging power series that allow easy and fast numerical evaluation. Explicit representation of wavelets facilitates, among other things, the study of the analytic properties of wavelets.

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  • 1. I. Daubechies, Ten Lectures on Wavelets, SIAM Publications, Soc. Indust. Appl. Math., Philadelphia (1992). MR 93e:42045
  • 2. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions, Vol. 1 McGraw-Hill, New York (1953). MR 15:419i
  • 3. H. Exton, Multiple Hypergeometric Functions And Applications, John Wiley & Sons, New York (1976). MR 54:10699
  • 4. I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York (1965). MR 33:5952
  • 5. E. Hernandez and G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, Florida (1996). MR 97i:42015
  • 6. H. Srivastava and H. Manocha, A Treatise on Generating functions, John Wiley & Sons, New York (1984). MR 85m:33016
  • 7. W.H. Young, On infinite integrals involving a generalization of the sine and cosine functions, Quart. J. Math., Vol. 4 (1912), pp. 161--177.
  • 8. G. Walter, Translation and dilation invariance in orthogonal wavelets, Appl. Comp. Harmonic Anal., Vol. 1 (1994), pp. 344--349. MR 96b:42040
  • 9. A. Zayed and G. Walter, Wavelets in Closed Forms, in Wavelet Transforms and Time-frequency Signal Analysis, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, MA, 2001, pp. 121-143. MR 2002c:42061

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Additional Information

Ahmed I. Zayed
Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614

Keywords: Orthonormal wavelets, bandlimited wavelets, multiresolution analysis, special functions, hypergeometric functions, Kamp\'e de F\'eriet functions
Received by editor(s): November 8, 2000
Published electronically: May 1, 2002
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society

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