Fractional Fischer decomposition, the ternary case

Transcription

1 Fractional Fischer decomposition, the ternary case Aurineide Fonseca Department of Mathematics, University of Aveiro, Portugal & Federal University of Piauí, Brazil MOIMA06-06/06/ Joint with P. Cerejeiras, M. Vajiac and N. Vieira

2 Introduction In the classic Clifford algebra setting, the construction of a monogenic function theory is based on the construction of a so-called Howe dual pair consisting of a Super-Lie algebra usually osp and a Spinor space; Super-Lie algebra osp is generated by three operators: the Dirac operator, the vector variable operator, and the so-called Euler operator or radial derivative; However, this construction fails in the present case; The principal reason is that the choice of osp is based on the preference of a SU symmetry of the classic Clifford algebras; In this talk we construct a so-called Fischer dual pair and introduce an inner product in the space of homogeneous polynomials with values in the generalized ternary Clifford algebra. We conclude with an algorithmic decomposition of homogeneous polynomials.

3 Outline Preliminaries Ternary Clifford Algebras Cl / d Generalized Fractional Calculus Fractional Fischer Decomposition Fractional Relations Fractional Fischer Decomposition Explicit Formulae Explicit Algorithm Sub-algebra in Cl /

4 Preliminaries Ternary Clifford Algebras Cl / d Ternary Clifford Algebras Cl / d Let {e,, e d } be the standard basis of the Euclidean vector space C d. The associated ternary Clifford algebra Cl / d is the free algebra generated by C d subject to the multiplication rule: e i =, e i e j = ωe j e i, for i < j, where ω = e iπ/. This multiplication satisfies [e i, e j, e k ] = e i e j e k + e i e k e j + e j e i e k + e j e k e i + e k e i e j + e k e j e i = 6δ ijk A vector space basis for Cl / d is given by the set e ν := e ν e ν d d where ν = ν,, ν d is an ordered d tuple with ν j = 0,,. Moreover, we obtain the following commutator relations: which lead to: e ν i i e µ j j = ω ν i µ j e µ j j e ν i i, i < j, e ν e µ = ω ν µ e ν+µ, where ν µ := ν d + ν d + + ν µ + ν d + ν d + + ν µ + + ν d µ d.

5 Preliminaries Ternary Clifford Algebras Cl / d Ternary Clifford Algebras Cl / d Therefore the ternary Clifford algebra Cl / d Cl / d = { w = ν has the form: w ν e ν, w ν C, ν = ν,, ν d, ν j = 0,, }, where we recall the complex scalars z C commute with the basis elements, i.e., ze j = e j z for all j =,..., d. The conjugation in this ternary Clifford algebra Cl / d : Cl / d Cl / d given by: w w = ν is, by definition, the automorphism w ν e ν, 4 where w ν denotes the usual complex conjugation and e ν = e ν e ν d d := e ν d d... e ν.

6 Preliminaries Ternary Clifford Algebras Cl / d Ternary Clifford Algebras Cl / d Following standard computations, we obtain the product between a basis element and its conjugate: e ν i i e ν j j e µ i i e µ j j = e ν i i e ν j j e µ j j e µ i i = ω µ i µ j ν j e ν i µ i i e ν j µ j j, i < j, 5 which leads to: e ν e µ = ω ν µ e ν µ, 6 where ν µ := d k=0 µ d k ν d k µ d k + µ d k + + µ. The structure of this algebra will allow us to consider the polynomial operators of the form X = d k= k e k.

7 Preliminaries Generalized Fractional Calculus Generalized Fractional Calculus Definition Let the function ϕλ = k=0 ϕ k λ k, be an entire function with order ρ > 0 and degree σ > 0. We define the linear operator D ϕ, which acts on powers of z n as D ϕz 0 := 0, D ϕz n := ϕ n z n, n =,. ϕn We call D ϕ the fractional derivative associated to ϕ. The operation f z = a k z k k=0 D ϕ Dϕf z = k= a k ϕ k ϕ k z k 7 is said to be the Gelfond-Leontiev G-L operator of generalized differentiation with respect to the function ϕ, and the corresponding G-L integration operator is I ϕf z = k=0 a k ϕ k+ ϕ k z k+. 8 Remark From the theory of entire functions, the conditions required for ϕ should be given by k ϕ lim sup k k ϕ =. However, we assume that the limit exits and that k k ϕ lim k k ϕ =. By the Cauchy-Hadamard formula, both series, 7 and 8, have the k same radius of convergence R > 0.

8 Preliminaries Generalized Fractional Calculus Generalized Fractional Calculus Example Let ϕλ be a Mittag-Leffler function of the for ϕλ = E ρ,µ λ = λ k, ρ > 0, µ C, k=0 Γ µ + k ρ with Reµ > 0, or for simplicity: with real ρ > 0, µ > 0. Then ϕ k λ = Γ µ+ ρ k and operators 7, 8 turn into the so-called Dzrbashjan-Gelfond-Leontiev D-G-L operators of differentiation and integration: Γ µ + k ρ D ρ,µf z = a k z k, I ρ,µf z = a k k= Γ µ + k ρ k=0 Γ µ + k ρ z k+. 9 Γ µ + k+ ρ

9 Preliminaries Generalized Fractional Calculus Generalized Fractional Calculus Kiryakova studied the connections between the D-G-L operators 9 and the so-called Erdélyi-Kober E-K fractional integrals and derivatives. Kiryakova presented transmutation operators relating Riemann-Liouville R-L fractional integrals R ρ and D-G-L generalized integrations L ρ,, L ρ,µ and represented by E-K operators. During the talk we will consider the ternary Dirac operator D, such that D = d j= e i D / i, where D / i represents the G-L generalized derivative 7 with respect to the coordinate x i. In analogous to the euclidian case a Cl / d -valued function f is called ternary left-monogenic if it satisfies Df = 0 on Ω resp. ternary right-monogenic if it satisfies fd = 0 on Ω.

10 Fractional Fischer Decomposition Fractionall Relations Fractional Relations We will first analyze how the differential operators act on the variables x i D / i k i l = ϕl, l D / i k i l [] Therefore at x = 0 we obtain: = ϕl, l ϕl, l D / i k i l [] = = ϕl, l ϕl, l ϕl k +, l k i l k. 0 D / i k i l x=0 = { 0, if k l; ϕl, l ϕ, 0 if k = l; and we write Φ l = ϕl, l ϕ, 0.

11 Fractional Fischer Decomposition Fractionall Relations Fractional Relations Fractional relations We will consider the following fractional relations [ ] D / i, j l r = = D / i j j D / i r 0, if i j; l ϕl, 0 r, if i = j i r; l ϕ D l +, l r, if i = j = r, with l N, i, j, r =,..., d and ϕ D l +, l = ϕl +, l ϕl, l. l Example For the case of Mittag-Leffler functions we have that ϕa, b = Γ+ a Γ+ b.

12 Fractional Fischer Decomposition Fractional Fischer Decomposition Fractional Fischer Decomposition Fischer Decomposition on polynomials Any homogeneous polynomial with coefficients in our algebra can be written as: P k X = X n a n, a n Cl / d, n N d 0 : n =k with n N d 0, k = n = n n d denoting the degree of the polynomial, X n = n d n d and a n Cl / d has the form a n = ν an,ν eν. The Fischer inner product of two fractional homogeneous polynomials P and Q of degree k is given by [ P, Q k = Sc P QX], x=0 where P is a differential operator obtained by replacing in the polynomial P each variable j by its corresponding fractional derivative D / j. From we immediately get that for any polynomial P l of homogeneity l and any polynomial Q l of homogeneity l it holds X P l, Q l l = P l, DQ l l. 4

13 Fractional Fischer Decomposition Fractional Fischer Decomposition Fractional Fischer Decomposition Theorem For each l N 0 we have Π l = M l + X Π l, where Π l denotes the space of fractional homogeneous polynomials of degree l and M l denotes the space of fractional monogenic homogeneous polynomials of degree l. Moreover, the subspaces M k and X Π l are orthogonal with respect to the Fischer inner product. In consequence, we obtain the fractional Fischer decomposition with respect to the fractional Dirac operator D. Theorem Let P l be a fractional homogeneous polynomial of degree l. Then P l = M l + X M l + X M l X l M 0, 5 where each M j denotes the fractional monogenic polynomial of degree j. More specifically, M 0 Π 0, and M l {u Π l : Du = 0}. The spaces represented in 5 are orthogonal to each other with respect to the Fischer inner product.

14 Explicit Formulae Explicit Algorithm Explicit Algorithm From the Fischer decomposition 5 we obtain: dimm l = dimπ l dimπ l, with the dimension of the space of fractional homogeneous polynomials of degree l given by This leads to the following theorem: dimπ l = l + d! l! d!. Theorem The space of fractional homogeneous monogenic polynomials of degree l has dimension l + d! ll + d! dimm l = = l! d! l + d! l! d!.

15 Explicit Formulae Explicit Algorithm Explicit Algorithm As we have seen before, any homogeneous polynomial of degree k = n with coefficients in Cl / d can be written as P k X = X n a n, a n Cl / d, n N d 0 : n =k with k = n = n n d denoting the degree of the polynomial. We now check under which conditions we have DP k = 0, i.e., 0 = D X n a n n N d 0 : n =k = = DX n a n n N d 0 : n =k d e j ϕn j, n j j X n a n j= n N d 0 : n =k 6

16 Explicit Formulae Explicit Algorithm Explicit Algorithm The last equality leads to the following theorem Theorem Equation 6 is equivalent to the following linear system M A = 0, 7 where A = [a l,...,l d ] dimπl, 0 = [0] dimπl are vectors, and M is the matrix with entrances given by [ ] M = M k,...,k d,l,...,l d, dimπ l dimπ l { ei ϕl M k,...,k d,l,...,l d = i, k i, k i = l i k j = l j i j 0, others cases.

17 Explicit Formulae Explicit Algorithm Explicit Algorithm Let us now indicated a possible ordering for the rows of system 7. In order to do that, let us consider the ordered set L = {L i = l i,..., l i d : Li = l = l i l i d, i =,..., dimπ l }, where the relation order is given by with L i > L i+ l i,..., l i d > l i+,..., l i+ d l i i l i... l i d > l i+ l i+... l i+ d l k l k... l k d := l k 0d + l k 0d l k d 00. Applying this ordering we get the following corollary. Corollary The matrix M has the following structure: M = M M, where the sub-matrix M = [m ij ] dimπ l dimπ l is an upper triangular matrix with entrances given by

18 Explicit Formulae Explicit Algorithm Explicit Algorithm Corollary M = e ϕl, l e ϕ, 0 e 4 ϕ, 0 e d ϕ, e ϕl, l e ϕ, e d ϕ, 0 e d ϕ, e ϕ, 0, and the sub-matrix M = [m ij ] dimπ l dimπ l has its entrances given by M = e ϕl, l e ϕ, 0 e d ϕ, e ϕl, l e d ϕ, 0 e d ϕ, e ϕ, 0 e d ϕl, l..

19 Explicit Formulae Explicit Algorithm Explicit Algorithm For the resolution of system 7 we implement the following algorithm to obtain the coefficients. Since M is upper triangular matrix we can treat a dimπl +,..., a dimπl as free parameters and obtain the following formula for the coefficients. Let the entry n, n of M correspond to the index [k, k,..., k d, k +, k,..., k d ], then [ d ] a n = e ϕk +, k e j ϕk j +, k j a k,...,k j +,...,k d, 8 j= where a n M n,n M k,l...,k d,k +,k,...,k d.

20 Explicit Formulae Sub-algebra in Cl / Sub-algebra in Cl / For the implementation we use the following two matrix representation for this space: E = , E = 0 ω ω 0 0, E = ω ω. E = , E = 0 0 ω ω 0, E = ω ω, E E = 0 0 ω ω 0, E E = 0 ω ω 0 0. This representation determines a sub-algebra in our space, yielding the extra condition E E = E. We need to determine a complementary sub-algebra. A second representation is found by finding the two other matrices that cube to I that are obtained from the unit vectors, 0, 0, 0,, 0 and 0, 0, and defining: E = ω ω, J = , J =

21 Explicit Formulae Sub-algebra in Cl / Sub-algebra in Cl / Example To illustrate the structure of M and A consider the case l =, and considering the Mittag-Leffler Γ+ a function, i.e. ϕa, b = Γ+ b. Taking into account Corollary 6, the vector A and the matrixes M, M take the form A T = a,0,0 a,,0 a,0, a,,0 a,, a,0, a 0,,0 a 0,, a 0,, a 0,0, = a a a a 4 a 5 a 6 a 7 a 8 a 9 a 0

22 Explicit Formulae Sub-algebra in Cl / Sub-algebra in Cl / Example cont. M = E ϕ, E ϕ, 0 E ϕ, E ϕ, 0 E ϕ, E ϕ, E ϕ, 0 E ϕ, 0 E ϕ, E ϕ, E ϕ, E ϕ, 0 M = E ϕ, E ϕ, E ϕ, E ϕ, E ϕ, 0 E ϕ, The columns of the matrix M are associated, respectively, to the last four elements of the matrix A.

23 Explicit Formulae Sub-algebra in Cl / Sub-algebra in Cl / Example cont. If we fix a 6, a 7, a 8, a 9 we can obtain, via formula 8, the remaining elements of the matrix A: a = E ϕ, [ ] E ϕ, 0 a,,0 + E ϕ, 0 a,0,, a = E ϕ, [ ] E ϕ, a,,0 + E ϕ, 0 a,,, a = E ϕ, [ ] E ϕ, 0 a,, + E ϕ, a,0,, a 4 = E ϕ, [ ] 0 E ϕ, a 0,,0 + E ϕ, 0 a 0,,, a 5 = E ϕ, [ ] 0 E ϕ, a 0,, + E ϕ, a 0,,, a 6 = E ϕ, [ ] 0 E ϕ, 0 a 0,, + E ϕ, a 0,0,. and therefore we solve system 7.

24 Explicit Formulae Sub-algebra in Cl / Sub-algebra in Cl / Example cont. For convenience of the reader we also give the basic monogenic polynomials for M and M, respectively V, V, V, V, V, = E +, = E E +. = E Γ 7 Γ E 5 x / = Γ 5 Γ E 7 EE +, = ω E E Γ 7 Γ 5 E E + +., E

25 Explicit Formulae Sub-algebra in Cl / Sub-algebra in Cl / Example cont. We can use the previous conclusions to obtain the four polynomials which are the basis for the space of fractional homogeneous monogenic polynomials M V, V, V, V, 4 = 7 + 8π = E Γ 7 Γ E 5 = ω E E Γ 7 Γ E 5 E E 7 E +, 8π + E E E +, = 7 + 8π 7 8π E E E E +, ω E E +.

26 Explicit Formulae Sub-algebra in Cl / The End! Thank you for your attention! Acknowledgement: This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology FCT Fundação para a Ciência e a Tecnologia, within project UID/MAT/ 046/0. Also, the author would like to thank the Organizers of MOIMA for their support.