What is nowadays called (classic) Clifford analysis consists in the establishment of a function theory for functions belonging to the kernel of the Dirac operator. While such functions can very well describe problems of a particle with internal SU(2)-symmetries, higher order symmetries are beyond this theory. Although many modifications (such as Yang-Mills theory) were suggested over the years they could not address the principal problem, the need of a n-fold factorization of the d’Alembert operator. In this paper we present the basic tools of a fractional function theory in higher dimensions, for the transport operator (alpha = 1/2 ), by means of a fractional correspondence to the Weyl relations via fractional Riemann-Liouville derivatives. A Fischer decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers are constructed

Vieira, Nelson

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20th International Conference on the Application of ComputerScience and Mathematics in Architecture and Civil EngineeringK. Gu¨rlebeck and T. Lahmer (eds.)Weimar, Germany, 20-22 July 2015SOME RESULTS IN FRACTIONAL CLIFFORD ANALYSISN. Vieira∗∗CIDMA - Center for Research and Development in Mathematics and ApplicationsDepartment of Mathematics, University of Aveiro, Campus Universita´rio de Santiago,3810-193 Aveiro, Portugal.E-mail: nloureirovieira@gmail.comKeywords: Fractional monogenic polynomials, Fischer decomposition, Fractional Dirac oper-ator, Riemann-Liouville fractional derivative, Stationary transport operator.Abstract. What is nowadays called (classic) Clifford analysis consists in the establishmentof a function theory for functions belonging to the kernel of the Dirac operator. While suchfunctions can very well describe problems of a particle with internal SU(2)-symmetries, higherorder symmetries are beyond this theory. Although many modifications (such as Yang-Millstheory) were suggested over the years they could not address the principal problem, the need ofa n-fold factorization of the d’Alembert operator.In this paper we present the basic tools of a fractional function theory in higher dimensions,for the transport operator (α = 12), by means of a fractional correspondence to the Weyl rela-tions via fractional Riemann-Liouville derivatives. A Fischer decomposition, fractional Eulerand Gamma operators, monogenic projection, and basic fractional homogeneous powers areconstructed.2121 INTRODUCTIONIn the last decades the interest in fractional calculus increased substantially. This fact isdue to on the one hand different problems can be considered in the framework of fractionalderivatives like, for example, in optics and quantum mechanics, and on the other hand fractionalcalculus gives us a new degree of freedom which can be used for more complete characterizationof an object or as an additional encoding parameter.Over the last decades F. Sommen and his collaborators developed a method for establishinga higher dimension function theory based on the so-called Weyl relations [1, 3, 2]. In morerestrictive settings it is nowadays called Howe dual pair technique (see [6]). Its focal point isthe construction of an operator algebra (classically osp(1|2)) and the resulting Fischer decom-position.The aim of this paper is to present a Fischer decomposition, when considering the fractionalDirac operator defined via Riemann-Liouville derivatives, where the fractional parameter isequal to 12(which leads to the case of the stationary transport operator). The results presentedhere correspond to a restriction of correspondent ones presented in [7] for the particular case ofα = 12.In the Preliminaries we recall some basic facts about Clifford analysis and fractional cal-culus. In Section 3, we introduce the corresponding Weyl relations for this fractional settingand the notion of a fractional homogeneous polynomial. In the same section we present thefractional correspondence to the Fischer decomposition. In the final section we construct theprojection of a given fractional homogeneous polynomial into the space of fractional homo-geneous monogenic polynomials. We also calculate the dimension of the space of fractionalhomogeneous monogenic polynomials.2 PRELIMINARIESWe consider the d-dimensional vector spaceRd endowed with an orthonormal basis {e1, · · · ,ed}. We define the universal real Clifford algebra R0,d as the 2d-dimensional associative algebrawhich obeys the multiplication rules eiej + ejei = −2δi,j. A vector space basis for R0,d isgenerated by the elements e0 = 1 and eA = eh1 · · · ehk , where A = {h1, . . . , hk} ⊂ M ={1, . . . , d}, for 1 ≤ h1 < · · · < hk ≤ d. An important subspace of the real Clifford algebraR0,d is the so-called space of paravectors Rd1 = R⊕Rd, being the sum of scalars and vectors.An important property of algebra R0,d is that each non-zero vector x ∈ Rd1 has a multiplicativeinverse given by x||x||2 . Now, we introduce the complexified Clifford algebra Cd as the tensorproductC⊗ R0,d ={w =∑AwAeA, wA ∈ C, A ⊂M},where the imaginary unit i of C commutes with the basis elements, i.e., iej = eji for allj = 1, . . . , d.An Cd−valued function f over Ω ⊂ Rd1 has representation f =∑A eAfA, with componentsfA : Ω → C. Properties such as continuity are understood component-wisely. Next, we recallthe Euclidean Dirac operator D =∑dj=1 ej ∂xj , which factorizes the d-dimensional EuclideanLaplacian, i.e., D2 = −∆ = −∑dj=1 ∂x2j . A Cd-valued function f is called left-monogenic if213it satisfies Du = 0 on Ω (resp. right-monogenic if it satisfies uD = 0 on Ω). For more detailsabout Clifford algebras and monogenic function we refer [2].The most widely known definition of the fractional derivative is the so-called Riemann-Liouville definition:(Dαa+f) (x) =(ddx)n1Γ(n− α)∫ xaf(t)(x− t)α−n+1 dt, n = [α] + 1, a, x > 0. (1)where [α] means the integer part of α. In [4] fractional derivative (1) was successfully appliedin the definition of the fractional correspondent of the Dirac operator in the context of Cliffordanalysis. In fact, for the particular case of α = 12, the fractional Dirac operator correspondsto D =∑dj=1 ej Dj =∑dj=1 ej (Dj + Yj), where Dj = ∂xj and Yj =12(ξj−xj) with ξ =(ξ1, . . . , ξd) the observer time vector. Moreover, we have that DD = D with D the EuclidianDirac operator, i.e., the stationary transport operator. ACn-valued function f is called fractionalleft-monogenic if it satisfies Du = 0 on Ω (resp. fractional right-monogenic if it satisfiesuD = 0 on Ω). We observe that due to the definition of D we have thatD(d∏i=1(ξi − xi) 12)= 0, (2)i.e.,∏di=1(ξi − xi)12 is a fractional monogenic function. The fractional power (ξi − xi) 12 corre-sponds to (ξj − xj) 12 if ξj ≥ xj , or (xj − ξj) 12 i if ξj < xj , with j = 0, 1, . . . , d. From now untilthe end of the paper, we consider paravectors of the form x = x0 + x, where x =∑dj=1 ej xjwith xj =ξj−xj2.3 WEYL RELATIONS AND FRACTIONAL FISCHER DECOMPOSITIONThe aim of this section is to provide the basic tools for a function theory for the fractionalDirac operator defined via fractional Riemann-Liouville derivatives for the particular case ofα = 12.3.1 Fractional Weyl relationsNow we introduce the fractional correspondence of the classical Euler and Gamma opera-tors. Furthermore, we show that the two natural operators D and x, considered as odd elements,generate a finite-dimensional Lie superalgebra in the algebra of endomorphisms generated bythe partial fractional Riemann-Liouville derivatives, the basic vector variables xj (seen as mul-tiplication operators), and the basis of the Clifford algebra ej .In order to obtain our results, we use some standard technique in higher dimensions, namelywe study the commutator and the anti-commutator between x and D. We start proposing thefollowing fractional Weyl relations[ Di,xi ] = Di xi − xi Di = −12, (3)with i = 1, . . . , d. This leads to the following relations for x and D:{D,x} = Dx + xD = −2E+ d2, [x,D] = xD−Dx = −2Γ− d2, (4)214where E, Γ are, respectively, the fractional Euler and Gamma operators of order 12, and have thefollowing expressionsE =d∑i=1xi Dj, Γ =∑i<jeiej (xi Dj −Di xj) . (5)From (5) we derive[x,E] =12x, [D,E] = −12D, (6)which allow us to conclude that we have a finite dimensional Lie superalgebra generated by xand D, isomorphic to osp(1|2). Now we introduce the definition of fractional homogeneity ofa polynomial by means of the fractional Euler operator.Definition 3.1 A polynomial Pl is called fractional homogeneous of degree l ∈ N0, if and onlyif EPl = − l2 Pl.We observe that from the previous definition the basic fractional homogeneous powers are givenby∏dj=1 xβjj , with l = |β| = β1 + . . . + βd. In combination with the first relation in (6) thisdefinition also implies that the multiplication of a fractional homogeneous polynomial of degreel by x, results in a fractional homogeneous polynomial of degree l+ 1, and thus may be seen asa raising operator. Moreover, we can also ensure that for a fractional homogeneous polynomialPl of degree l, DPl is a fractional homogeneous polynomial of degree l − 1. Furthermore,Weyl’s relations (3) enable us to construct fractional homogeneous polynomials, recursively.3.2 Fractional Fischer decompositionA fractional Fischer inner product of two fractional homogeneous polynomials P and Qwould have the following form〈P (x), Q(x)〉 = Sc[P (∂x) Q(x)], (7)where ∂x represents Dj , and P (∂x) is a differential operator obtained by replacing in the poly-nomial P each variable xj by the corresponding fractional derivative, i.e. Dj +Yj . From (7) wehave that for any polynomial Pl−1 of homogeneity l−1 and any polynomial Ql of homogeneityl the relation 〈x Pl−1, Ql〉 = 〈Pl−1,DQl〉. This fact allows us to obtain the following result:Theorem 3.2 For each l ∈ N0 we have Πl = Ml + x Πl−1, where Πl denotes the spaceof fractional homogeneous polynomials of degree l and Ml denotes the space os fractionalmonogenic homogeneous polynomials of degree l. Moreover, the subspacesMk and x Πl−1 areorthogonal with respect to the Fischer inner product (7).The proof is analogous to the proof of Theorem 3.5 in [7] but considering α = 12, and thereforewe omit it from the paper. As a result of the previous theorem we obtain the fractional Fischerdecomposition with respect to the fractional Dirac operator D.Theorem 3.3 Let Pl be a fractional homogeneous polynomial of degree l. ThenPl = Ml + x Ml−1 + x2 Ml−2 + . . .+ xl M0, (8)where each Mj denotes the fractional homogeneous monogenic polynomial of degree j. Morespecifically, M0 = P0 and Ml = {u ∈ Pl : Du = 0}.The spaces represented in (8) are orthogonal to each other with respect to the Fischer innerproduct (7). This a consequence of the construction of the fractional Euler operator E (see (5)),and in particular of (4).2153.3 Explicit formulaeHere we obtain an explicit formula for the projection piM(Pl) of a given fractional homo-geneous polynomial Pl into the space of fractional homogeneous monogenic polynomials. Westart with the following auxiliary result:Theorem 3.4 For any fractional homogeneous polynomial Pl and any positive integer s, wehave DxsPl = gs,lxs−1Pl + (−1)sxsDPl, where g2k,l = k and g2k+1,l = k + l + d2 .Proof: The proof follows, by induction and straightforward calculations, from the commuta-tion between D and xs using the relations Dx = −2E+ d2− x D and Ex = x E− 12x.Let us now compute an explicit form of the projection piM(Pl).Theorem 3.5 Consider the constants cj,l defined by c0,l = 1, and cj,l =(−1)j Γ( d2+l−1−[ j2 ])Γ( d2+1) Γ(1+2[j2 ]),where j = 1, . . . , l and [·] represents the integer part. Then the map piM given bypiM(Pl) := Pl + c1,l x D Pl + c2,l x2D2 Pl + . . .+ cl,l xlDl Plis the projection of the fractional homogeneous polynomial Pl into the space of fractional ho-mogeneous monogenic polynomials.Proof: Let us consider the linear combinationr = a0 Pl + a1 x D Pl + a2 x2 D2 Pl + . . .+ ak xl Dl Pl,with a0 = 1. If there are constants aj , j = 1, . . . , l, such that r ∈Ml, then r is equal to piM(Pl).Indeed, we know that Pl = Ml ⊕ xPl−1 and r = Pl + Ql−1, with Ql−1 =∑li=1 ai xi Di Pl.Applying Theorem 3.4, we get0 = D(piM(Pl))= D Pl + a1 D x D Pl + a2 D x2 D2 Pl + . . .+ al D xlDl Pl= (1 + a1 g1,l−1) D Pl + (−a1 + a2 g2,l−2)x D2 Pl + (a2 + a3 g3,l−3) x2 D3 Pl+ . . .+ ((−1)l−1al−1 + al gl,0) xl−1 Dl Pl.Hence if the relation (−1)j−1aj−1 + aigj,l−j = 0 holds for each j = 1, . . . , l, then the functionr is fractional monogenic. By induction we get aj =(−1)j Γ( d2+l−1−[ j2 ])Γ( d2+1) Γ(1+2[j2 ]).Theorem 3.6 Each polynomial Pl can be written in a unique way as Pl =∑lj=0 xj Ml−j(Pl),where Ml−j(Pl) = c′j∑jn=0 cj,l−n xn DnDl−j Pl with j = 0, . . . , l, and the coefficients c′j aredefined by c′j =(−1)j Γ( d2+l−1−[ j2 ])Γ( d2+1) Γ(1+2[j2 ]).The proof of this result is analogous to the proof of Theorem 3.9, for α = 12, in [7], and thereforewe omit it from the paper.Acknowledgement: This work was supported by Portuguese funds through the CIDMA -Center for Research and Development in Mathematics and Applications, and the PortugueseFoundation for Science and Technology (“FCT–Fundac¸a˜o para a Cieˆncia e a Tecnologia”),within project UID/MAT/ 0416/2013. N. Vieira was also supported by FCT via the FCT Re-searcher Program 2014 (Ref: IF/00271/2014).216REFERENCES[1] H. De Bie and F. Sommen: Fischer decompositions in superspace. Le Hung Son et al eds.Function spaces in complex and Clifford analysis, National University Publishers, Hanoi,2008.[2] R. Delanghe, F. Sommen and V. Souc˘ek: Clifford algebras and spinor-valued functions. Afunction theory for the Dirac operator. Mathematics and its Applications-Vol.53, KluwerAcademic Publishers, Dordrecht etc., 1992.[3] H. De Ridder, H. De Schepper, U. Ka¨hler and F. Sommen: Discrete function theory basedon skew Weyl relations. Proc. Am. Math. Soc., 138-No.9, 3241-3256, 2010.[4] R.A. El-Nabulsi: Fractional Dirac operators and deformed field theory on Clifford algebra.Chaos Solitons Fractals, 42-No.5, 2614-2622, 2009.[5] U. Ka¨hler and N. Vieira: Fractional Clifford analysis. S. Bernstein, U. Ka¨hler, I. Sabadiniand F. Sommen eds. Hypercomplex Analysis: New perspectives and applications, Trendsin Mathematics, Birkha¨user, Basel, 2014.[6] B. Ørsted, P. Somberg and V. Soucˇek: The Howe duality for the Dunkl version of theDirac operator. Adv. Appl. Clifford Algebr., 19-No.2, 403–415, 2009.[7] N. Vieira: Fischer Decomposition and Cauchy-Kovalevskaya extension in fractional Clif-ford analysis: the Riemann-Liouville case. Proc. Edinb. Math. Soc., II. Ser., in press.217

SOME RESULTS IN FRACTIONAL CLIFFORD ANALYSIS

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SOME RESULTS IN FRACTIONAL CLIFFORD ANALYSIS

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