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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**Ingrid Daubechies,*Ten lectures on wavelets*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR**1162107****2.**Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,*Higher transcendental functions. Vols. I, II*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR**0058756****3.**Harold Exton,*Multiple hypergeometric functions and applications*, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York-London-Sydney, 1976. Foreword by L. J. Slater; Mathematics & its Applications. MR**0422713****4.**I. S. Gradshteyn and I. M. Ryzhik,*Table of integrals, series, and products*, Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin. Translated from the Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey, Academic Press, New York-London, 1965. MR**0197789****5.**Eugenio Hernández and Guido Weiss,*A first course on wavelets*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. With a foreword by Yves Meyer. MR**1408902****6.**H. M. Srivastava and H. L. Manocha,*A treatise on generating functions*, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1984. MR**750112****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.**Gilbert G. Walter,*Translation and dilation invariance in orthogonal wavelets*, Appl. Comput. Harmon. Anal.**1**(1994), no. 4, 344–349. MR**1310657**, 10.1006/acha.1994.1020**9.**Ahmed I. Zayed and Gilbert G. Walter,*Wavelets in closed forms*, Wavelet transforms and time-frequency signal analysis, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2001, pp. 121–143. MR**1826011**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
42C40,
33C20,
42C15,
33E20

Retrieve articles in all journals with MSC (2000): 42C40, 33C20, 42C15, 33E20

Additional Information

**Ahmed I. Zayed**

Affiliation:
Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614

Email:
azayed@condor.depaul.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06690-X

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