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Some Ultraspheroidal Monogenic Clifford Gegenbauer Jacobi

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Some Ultraspheroidal Monogenic Clifford Gegenbauer Jacobi Some Ultraspheroidal Monogenic Clifford Gegenbauer Jacobi Polynomials and Associated Wavelets Sabrine Arfaoui Department of Informatics, Higher Institute of Applied Sciences and Technology of Mateur, Street of Tabarka, 7030 Mateur, Tunisia. and Research Unit of Algebra, Number Theory and Nonlinear Analysis UR11ES50, Faculty of Sciences, Monastir 5000, Tunisia. Anouar Ben Mabrouk Higher Institute of Applied Mathematics and Informatics, University of Kairouan, Street of Assad Ibn Alfourat, Kairouan 3100, Tunisia. and Research Unit of Algebra, Number Theory and Nonlinear Analysis UR11ES50, Faculty of Sciences, Monastir 5000, Tunisia. Abstract In the present paper, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of orthogonal polynomials are provided based on 2-parameters weight functions. Such classes englobe the well known ones arXiv:1704.03512v1 [math.CA] 6 Apr 2017 of Jacobi and Gegenbauer polynomials when relaxing one of the parameters. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula as well as Fourier-Plancherel rules have been proved. Key words: Clifford Gegenbauer Jacobi polynomials, Continuous Wavelet Transform, Clifford analysis, Clifford Fourier transform, Fourier-Plancherel. PACS: : 42B10, 44A15, 30G35. Email addresses: [email protected] (Sabrine Arfaoui), [email protected] (Anouar Ben Mabrouk). Preprint submitted to Elsevier 21 October 2018 1 Introduction Spheroidal functions or precisely spheroidal wave functions are issued from the wave equation ∇2w + k2w =0. By considering solutions of separated variables in an elliptic cylinder coordi- nates system, or the prolate or oblate spheroids, such solutions satisfy a second order ODE of the form (1 − t2)w′′ +2αtw′ +(β − γ2t2)w =0. for both radial and angular functions. In fact, prolate and oblate spheroidal coordinate systems are results of rotating the two-dimensional elliptic coordi- nate system, consisting of confocal ellipses and hyperbolas, about the major and minor axes of the ellipses. See [1], [22], [28], [36], [37][31]. This last equa- tion leads to special functions such as Bessel, Airy, ... and special polynomials such as Gegenbauer, Legendre, Chebyshev, .... This is a first idea behind the link between these functions and a first motivation of our work and its titling as spheroidal wavelets. Besides, spheroidal functions have been in the basis of modeling physical phenomena where the wave behaviour is pointed out such as radars, antennas, 3D-images, ... Recall also that Gegenbauer polynomials themselves are strongly related to spheroidal functions since their appearance and these are called ultraspheroidal polynomials. See [2], [11], [12], [21], [22], [26], [29], [33]. The use of wavelets in the analysis of functions is widespread especially in the last decades. Nowadays, wavelets are interesting and useful tools in many fields such as mathematics, quantum physics, electrical engineering, time/image pro- cessing, bio-signals, seismology, geology, ..... Wavelets have been created to meet a need in signal processing that is not well understood by Fourier theory. Classical Fourier analysis provides a global approach for signals as it replaces the analyzed function with a whole-space description (See (11) later). Wavelet analysis in contrast decomposes the signal in both time and frequency and describes it locally and globally, as the need. Wavelet analysis of a function f in the space of able analyzed functions (gen- erally L2) starts by convoluting it with local copy of a wavelet mother function ψ known as the analyzing wavelet relatively to 2-parameters; One real num- ber parameter a> 0 defines the dilation parameter or the scale and one space parameter b in the same space as the function f and ψ domains defines the translation parameter or the position. Such copy is denoted usually by ψa,b and is defined by − 1 x − b ψ (x)= a 2 ψ( ). (1) a,b a 2 To be a good candidate as a wavelet mother, an admissibility assumption on the function ψ is usually assumed. It states that +∞ |ψ(u)|2 Aψ = du < +∞, (2) −∞ |u| Z b where ψ is the Fourier transform of ψ. b The convolution of the analyzed function f with the copy ψa,b defines the so- called wavelet transform of f or exactly the Continuous Wavelet Transform (CWT) expressed by +∞ Ca,b(f)=< f,ψa,b >= f(x)ψa,b(x)dx. (3) −∞ Z Whenever the admissibility condition is fulfilled, the analyzed function f may be reconstructed in an L2 sense as 1 +∞ da f(x)= Ca,b(f)ψa,b(x) db, (4) R 2 Aψ Z Z0 a where the equality has to be understood in the L2-sense (See [15], [16]). This equality will be proved later in the present context of Clifford Gegenbauer- Jacobi type wavelets. Usually analyzing wavelets are related also to moments. The regularity of the analyzing wavelet ψ is related to a number of vanishing moments that should be satisfied +∞ xnψ(x)dx =0, n =0, 1, . , N. (5) −∞ Z Such a condition helps to analyze functions of some fixed regularity. In wavelet theory, the first result relating regularity to wavelet transforms is due to Jaf- fard (See [14], [15], [16]) and is stated as follows. Proposition 1 Let ψ be a Cr(Rm) function with all moments of order less than r vanishing and all derivatives of order less than r well localized. α m α • f ∈ C (R ) if and only if |Ca,b(f)| ≤ Ca for all b and 0 <a<< 1. α • If f ∈ C (x0), then for 0 <a<< 1 and |b − x0| ≤ 1/2, α α |b − x0| |Ca,b(f)| ≤ Ca 1+ . (6) a ! • If (6) holds and if f ∈ Cε(Rm) for an ε> 0, then there exists a polynomial P such that, if |x − x0| ≤ 1/2, α 2 |f(x) − P (x − x0)| ≤ C|x − x0| log . (7) |x − x0|! 3 More about regularity, admissibility, vanishing moments and wavelet proper- ties may be found in [10], [12], [18], [27], [32], [39]. It holds that wavelet theory on the real line and generally on Euclidian spaces has been extended in some cases of Clifford analysis. The classical wavelet the- ory can be constructed in the framework of Clifford analysis. Clifford analysis deals with so-called monogenic functions which are described as solutions of the Dirac operator and/or direct higher dimensional generalizations of holo- morphic functions in the complex plane. Clifford wavelets and the possibility to construct orthogonal wavelet bases and consequently multiresolution analyses associated has been the object of several works, but remain to be a fascinating subject of researches. In [3] a multiresolution analysis in the context of Clifford analysis has been provided. Clifford scaling functions, Clifford wavelets as well as related wavelet filters has been developed.and proved to be applicable in quantum mechanics. In [19] and [20], spheroidal wavelets leading to frames as well as multiresolution analysis have been developed. It was proved that spheroidal functions may induce good candidates characterized by localiza- tions in both frequency and space and thus lead to good wavelets. More facts about Clifford wavelets and discussions on possible associated multiresolution analyses may be found in [4], [5], [6], [7], [8], [9], [13]. Let Ω be an open subset of Rm or Rm+1 and f : Ω → A, where A is the real Clifford algebra Rm (or Cm). f may be written in the form f = fAeA (8) XA where the functions fA are R (or C)-valued and (eA)A is a suitable basis of A. Despite the fact that Clifford analysis generalizes the most important features of classical complex analysis, monogenic functions do not enjoy all properties of holomorphic functions of one complex variable. For instance, due to the non-commutativity of the Clifford algebras, the product of two monogenic functions is in general not monogenic. It is therefore natural to look for specific techniques to construct monogenic functions. See [4], [12], [32]. In the literature, there are several techniques available to generate monogenic functions such as the Cauchy-Kowalevski extension (CK-extension) which con- sists in finding a monogenic extension g∗ of an analytic function g defined on a given subset in Rm+1 of positive codimension. For analytic functions g on m+1 the plane {(x0, x) ∈ R , x0 = 0} the problem may be stated as follows: Find g∗ ∈ A such that ∗ ∗ Rm+1 ∗ ∂x0 g = −∂xg in and g (0, x)= g(x). (9) 4 A formal solution is ∞ (−x )k g∗(x , x) = exp(−x ∂ )g(x)= 0 ∂kg(x). (10) 0 0 x k! x kX=0 It may be proved that (10) is a monogenic extension of the function g in Rm+1. Moreover, by the uniqueness theorem for monogenic functions this extension is also unique. See [4], [12], [32], [39], [40] and the references therein. The organization of this paper is as follows: In section 2, a brief overview of some properties of the Clifford and Fourier analysis has been conducted. Sec- tion 3 is devoted to a review of the class of Gegenbauer-Jacobi polynomials in the framework of Clifford analysis. In section 4, some new classes of polyno- mials generalizing those of section 3 are developed by adapting 2-parameters weights and thus applied to introduce some new wavelets. Section 5 is devoted to the link and discussions about the present case and Legendre and Tcheby- shev polynomials as well as the role of the parameters α and β in the Clifford weight function applied here. We concluded afterward. 2 Clifford analysis revisited In this section we revisit some basic concepts that will be used later. Let f be in L1(Rm). Its Fourier transform denoted usually f or F(f) is given by 1 b f(η)= F(f)(η)= m exp(−ix.η)f(x)dx, (11) 2 Rm (2π) Z b where dx is the Lebesgues measure on Rm and x.η is the standard inner product of x and η in Rm.
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