Some Old Orthogonal Polynomials Revisited and Associated Wavelets: Two-Parameters Clifford-Jacobi Polynomials and Associ
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Some Old Orthogonal Polynomials Revisited and Associated Wavelets: Two-Parameters Clifford-Jacobi Polynomials and Associated Spheroidal Wavelets Sabrine Arfaoui1,2 · Anouar Ben Mabrouk2,3,4
Received: 25 September 2016 / Accepted: 9 December 2017 © Springer Science+Business Media B.V., part of Springer Nature 2017
Abstract In the present paper, new classes of wavelet functions are developed in the framework of Clifford analysis. Firstly, some classes of orthogonal polynomials are provided based on two-parameters weight functions generalizing the well known Jacobi and Gegenbauer classes when relaxing the parameters. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula as well as Fourier-Plancherel rule have been proved. Keywords Continuous wavelet transform · Clifford analysis · Clifford Fourier transform · Fourier-Plancherel
1 Introduction The application of wavelets in the analysis of functions is widespread especially in the last decades. Since their discovery, wavelets have been an essential, interesting and useful tool in quasi all fields of sciences such as pure mathematics, computational mathematics, mathematical physics, computational physics, quantum physics, electrical engineering, time/image processing, bio-signals, seismology, geology, . . . . For backgrounds and further informations on these topics we may refer to [1, 2, 11, 13–17, 22, 26–29, 32, 33].
B S. Arfaoui
[email protected] A. Ben Mabrouk [email protected]
1
Department of Informatics, Higher Institute of Applied Sciences and Technology of Mateur, Street of Tabarka, 7030 Mateur, Tunisia
2
Algebra, Number Theory and Nonlinear Analysis Lab UR11ES52, Department of Mathematics, Faculty of Sciences, University of Monastir, 5019 Monastir, Tunisia
3
Department of Mathematics, Higher Institute of Applied Mathematics and Informatics, University of Kairouan, 3100 Kairouan, Tunisia
4
Department of Mathematics, College of Sciences, University of Tabuk, Tabuk, Saudi Arabia
S. Arfaoui, A. Ben Mabrouk
Wavelets have been introduced since their discovery to fill the need in signal and image processing that is not well understood or solved by Fourier theory. Classical Fourier analysis provides a global approach by replacing the analyzed function (signal) with a whole-space description as 1 exp −ix, η f (x)dx. (1) f(η) = F (f )(η) = m (2π) 2 Rm See for instance [26, 31]. Wavelet analysis in contrast decomposes the function (signal) in both time and frequency domains and describes the source locally and globally, as the need. Wavelet analysis of a function f in the space of analyzed functions (generally L2 (Rm )) starts by convoluting it with local copies of a source function ψ known as the analyzing wavelet (wavelet mother) relatively to two-parameters; One real number parameter a > 0 known as the scale and one space parameter b in the same space as the function f and ψ domains known as the position. Such copy is denoted usually by ψa,b defined by x −b 1 ψa,b (x
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