Taylor series and orthogonality of the Octonion analytic functions

2001,21B(3):323-330

TAYLOR SERIES AND ORTHOGONALITY OF THE OCTONION ANALYTIC FUNCTIONS 1 Li Xingmin ( 4'*~

)

Department of Mathematics, Guangzhou Normal University, Guangzhou 510405, China

Peng Lizhong ( jj;;i. 'f> ) Department of Mathematics, Beijing Unive"sity, Beijing 100871, China

Abstract

The Taylor series of the Octonion analytic function is given. And the orthog-

onal formula for the Octonion analytic functions is also obtained. Key words

Octonion, O-analytic function, Taylor series, orthogonality

1991 MR Subject Classification

1

30G35, 17A35

Introduction

The only finite dimensional alternative division algebras over R are Real algebra R, Complex algebra C, Quaternion algebra H, Octonion algebra 0 with the embedding relations: R l:;:; C l:;:; H l:;:; O. Rand C are commutative and associative, H is associative but not commutative, while 0 is neither commutative nor associativel'J. Much earlier the great Swiss mathematician Fueter (a student of Hilbert) and his students developed Quaternion analysis up to 1950s[2,3] and it was a great achievement to develop the higher-dimensional analogue of complex analysis. Another important direction to develop the higher-dimensional analogue of complex analysis is the so-called Clifford analysis. As a common generalization of Grassmann's exterior algebra and Hamilton's quaternions, the Clifford algebra An was constructed by, W.K. Clifford in 1878[4]. It has been intensively

studied since then and A o = R, Al = C, A 2 = H but A 3 i:- 0, because the Clifford algebra An are associative and are not division algebras (n 2: 3), while the octonions are division algebra but not associative. There are many important applications in mathematics and physics of the Clifford algebras, and there have been many remarkable research papers and excellent books (see [ 5, 6, 7,

8, 9]). The Octonion algebra 0 was discovered independently by J. J. Graves in 1843 and A. Cayley in 1845. It is important in algebra and physics, because the automorphism group of the non-associative algebra 0 is a compact Lie group whose Lie algebra is the 14-dimensional simple algebra of type G 2 in Cartan's classification. Also, its action of the sphere 56 is transitive 1 Received

May 4, 1999; revised May 16, 2000.

19631080 and 69735020).

The research supported by NNSF of China (Grant No.

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ACTA MATHEMATIC A SCIENTIA

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with isotropy group SU(3). So S6 ~ Gz/SU(3). In theoretical physics SU(3) is the group of internal symmetries of various families of particles (quarks, baryons, mesons etc.) Recently, Octonions are used in antisymmetric tensor gauge fields[lO]. But nearly nothing have been done about the Octonion analysis since the Octonion was discovered. In the papers [11, 12, 13], we gave the Cauchy integral formulas on the Octonions, which are the key step in the development of the Octonion analysis, the 3-line theorems, which is the analogue of the Hadamard 3-circle theorem in the complex analysis. Also, we found that the Stein-Weiss conjugate harmonic function is the Octonion analytic functionlv'l. And this was the first time to study analysis problems systematically on the Octonions. In this paper, we obtain the Taylor theorem on the Octonions, it is the key theorem of the Octonion analysis. By the Taylor theorem we can consider the Laurent expansion of the Octonion analytic functions, and pointwise singularities theory. In order to develop the £2 theory on the Octonions, the inner product should be defined. But, because of the non-associativity of the Octonions, it seems impossible. At least we still have no idea to define the inner product in the Octonion functions. In this paper, we obtain the orthogonal formula in the Octonion analytic functions, maybe it is helpful to develop the £2 theory on the Octonions. And we will consider it later.

2

Preliminaries The Octonion algebra, we denote it by 0, as a vector space over F (F

=R

or C), is an

alternative, non-associative division algebra with the basic Octonionic units: eo, ell ... , e6, e7 , where eo is the unit element in 0, satisfying 2_

eo - eo, eo eo

= eoeo,

a

= 0,1,2"",7,

where

DO f3

={

I if

°if

a

= {3,

a

f. {3.

The constants 1/Jof31' totally antisymmetric in a, {3, '"Y, are non-zero and equal the unity for the seven combinations: (1,2,3),(1,4,5),(2,4,6),(3,4,7),(2,5,7),(6, 1,7,),(5,3,6).

°

For the multiplication table, see [1, 15]. The basic elements of can be written as 1 = eo, ell e2, ele2, e4; ele4, e2e4, (ele2)e4. And any Octonion number, is of the form x L~ Xkek = (xoeo + xlel + X2e2 + X3e3) + (X4eO + xSel + X6e2 + x7e3)e4, Xj E F, j = 0,1,···,7. Let a = L~ akek, b = L~ bkek. We consider the product of the two Octonion numbers abo Putting

a

= ao + A, b = bo + B, then ab = aobo + aoB + boA + AB, Let Aij

= det

a i aj] [ bi bj

i,j = 1,2"",7.

=

Li & Peng: TAYLOR SERlES AND ORTHOGONALITY OF FUNCTIONS

No.3

325

Then

AB = -A· B + el(A 2,3 + A 4,5 - A6,7) + e2(-AI,3 + A 4,6 + A 5,7) +e3(A I,2 + A 4,7 - A 5,6) + e4(-A I,5 - A 2,6 - A 3,7) +e5(A I,4 - A Z,7 + A 3,6) + e6(A I,7 + A Z,4 - A 3,5) +e7( -A I,6 + A 2,5 + A 3,4). So, by using the symbol in [15], AB = -A· B + A x B, and (A x B) . A = 0, (A x B) .B = 0, All B {:=:> A x B = 0, A x B = -B x i.. Although the octonions do not satisfy the associative law, we still have: for all z , y, z E 0,

[x, z, y] = 0 = [y, x, x], [x, z , y] = 0 = [y, x, x] and [z, y, z] = [y, z , x] = [z, z, y], [x,z,y] = -[z,x,y], [y,x,z] = -[y,z,x], [y,x,z] = -[z,x,y], where [x,y,z] = (xy)z - x(yz) is called the associator of z , y and z . Also, the Octonions obey some weakened associative laws, such as the so-called R. Moufang identities: (uvu)x = u(v(ux)), x(uvu) = ((xu)v)u, u(xy)u = (ux)(yu)[I). Now, in what follows, we assume that F = R. And for each x E 0, its conjugate is defined: x = L~ Xkek, where eo = eo, ej = -ej, j = 1,2,···7, and calling Xo the real part 2 of z, Then ejej = ej ej, \if i,j = 1,2"",7. and xx = xx = L~ x; =: Ix1 • So if 0:3 x i- 0, X-I = i.e. 0 is a division algebra. And we have that 0 = H EB He4' H = CEBCel. Let n be an open connected set in R S , f : n ----> 0, f(x) = L~ek!k(x). The Dirac D-operator and its adjoint D are the first-order systems of differential operators on Coo(0,0) defined by

W.

D=2:>k axk ' D=L:>k axk ' o 7

/)

_

7

/)

0

Definition[ll) a function f in Coo(n,O) is said to be left (right) O-analytic on 0 when ,,7.!!L ( .!!L ) D f = L.."o ek OXk = 0 f D = L.."o OXk ek = 0 . Since

,,7

_

DD

_

7

= DD = 6 s = Lo

/)2

-2' /)x k

the real-valued components of any left (right) O-analytic function are always harmonic. The concept of a O-analytic function is a generalization of a H-analytic function defined by F'ueter[2,3).

3

Taylor Series

Let Sk(X) denote a real-valued spherical harmonic of order k, i.e. Sk(X) is a real valued homogeneous polynomial of degree k satisfying Laplace's equation. Let x = rw, w E S7, Then

Sk(X) = r k Sk(W), Sk(W) is called a surface spherical harmonic of order k. If H k (S7, R) denotes the space of all real-valued surface spherical harmonics of order k, then dimH k (S 7, R)

= N(k) =

(2k+6)r(k+6) r(k+l)r(7)'

{ 1,

= M(k) + M(k -

1).

k

~

1,

k=O

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Where M (k) denotes the number of coefficients in a homogeneous polynomial of order k in 7 variables[5J. We call a homogeneous polynomial of degree k, Pk

= I:~ ejPk,j(X),

x E 0, is a

left spherical a-analytic of order k if DPk = O. where Pk,j is a homogeneous polynomial of degree k. Clearly, each left spherical a-analytic function is harmonic of the same order. Let p k (L) denote the vector space of all the left spherical a-analytic functions. We want to construct a b~is for p k (L). Let ZI = Xleo-xoe" I = 1,2, ···,7. Then it is easy to verify that: DZI = 0, 1= 1,2, ... ,7.

f!

Now, Vk EN, Ij E {I, 2,,,,, 7}, let Vo(x) = eo, Vi ,12·" lk( X) = I:ll'(I ,h".,l k ) ZI , ZI 2)ZI.)"· Zl k ) , where the sum runs over all distinguishable permutations of all of (l1,12," . ,ld. And

hereafter we always assume that: ZI ,Z12)ZI.)",ZIJ = (",((ZI1Z1 2)ZI.)",ZIJ. Lemma 1 The polynomials V" 12,,.lk( X) are both the .left and the right spherical 0analytic function of order k. Proof It is obvious that the polynomials Vi ,12." lk(X) are homogeneous of order k. Next we have to prove that for all x E R 8 , DV" 12.,,l k (X) = O. Clearly, k = 1 is true. Now assume that k > 1, we have 1

DV,, 12,,·lk(X)

= k! 1 k!

7

L

LejOXjZI,Z12)ZI.)·ooZlk)

L

[eoOXOZl1Z12)ZI,),~,Zlk)+ LejOXjZI,Z12)ZI,)",Zlk)]

ll'(I 2 ,,,.,lk) 0 "1

ll'(I , ,12,,,.,l k

7

1

)

7

= O. An analogous computation shows that Remark Denote V 1k1k-,·"I

we still have DV I,12···lk(X) Lemma 2

,(X)

= ~!

V" 12.,,l (X)D k

L ll'(I , ,12,''',1 k)

= O.

(ZI 1(Zz,{oo. (Zl k,

= 0 = V I, 12·"lk(X) D.

Any left spherical a-analytic function of order k may be written as

where the sum runs over all possible combinations (1 2 , 12 , ••• lk) of k elements out of {I, 2, ... , 7}, repetitions being allowed.

Li & Peng: TAYLOR SERIES AND ORTHOGONALITY OF FUNCTIONS

No.3

327

Proof Pk(x) is of the form

Pk(X) =

2:: Caxgox~I .. ·x~r, lal=k

where

c.;

E O. We can check that 7

kPk(x)

= XOOxoPk(X) + 2:: XjOx;Pk(X). 1

By induction we can show that:

Pk(X)

=

2:: (1"1,, ... 1. )

VIII"""(X)OXI I'" OXI. Pk(X).

Remark If Pk(X) is a spherical right O-analytic function, with ViII, ...,.(X) in place of V'l"""'(x) ,then the lemma is still true. Lemma 3 {V'l"""': (h,12,···,lk) C {1,2,···7}} is a basis of pk(L), and dimpk(L)

=7M(k).

The proof of the lemma is similar with [5], so it is omitted. Assume that the function 1 is left O-analytic in an open set nCR 8 , without loss of

generality it may be supposed that n contains the origin. Let 1 = L::~ Ikek. Then l» is a harmonic in n. So there exists an open neighbourhood A of the origin in which it may be developed into its Taylor series

which together with all its derived series, all of them considered as a multiple power series in the real variables xo, X l , " ' , X7, converges normally in A, i.e. if j denotes a single summation index the terms of the multiple power series being ordered in an arbitrary manner, then for each compact subset K C A, J, 1 '~ " sup; I-xI k' I ... x, • 0XII ... 0Xl. IA(O)I

j=J 1 xEK

as inf(h, h)

---+ 00.

---+

•

Hence for the function

1 itself one gets

and also the series and its derived series converge normally in A. So

= 2:: Pd(x), 00

I(x)

o

where Pk ( :r) st.ands for the homogeneous polynomial of degree k.

0'

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ACTA MATHEMATIC A SCIENTIA

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and L:~ sUPxEK 1Pk/(x)1 is a convergent numerical series in any compact subset K C A. And we can easily show that: Lemma 4

If I is left 0 analytic in

n containing the ori~in, then for any

kEN, Pkl(x)

is a left spherical O-analytic function of order k. By the above lemmas, we can get Theorem 1 (Taylor type) If I is left O-analytic in an open set nCR 8 containing' the origin, then there exists an open neighbourhood A in which I can be developed into a normally convergent series of left spherical O-analytic functions:

L

00

I(x) =~)

VI1'2'"'kox'l''' ox,J(O))

o (ll,12,"',lk)

any derived series being normally convergent in A to the corresponding derivative of I. Lemma 5 (Weierstrass type)[ll] Let {fj}jEN be a sequence of left O-analytic functions in n. If for each compact set Ken and for any such that

€

> 0, there exists a natural number N(€, K),

sup I/i(X) - Ij(x)1 < e,

xEK

whenever i,j > N(€, K). Then there exists a function

I

n such

in

that

(i) DI = 0; (ii) the sequence {013/j hEN converges uniformly on the compact subsets of any multi-index (3 E N 8 . Using Lemma 5 and the method in [5], we also have Theorem 2 If the series of left spherical O-analytic functions 00

00

LPkl(x) o

= L( 0

L

(ll,1 2,,,.,l k)

n to 013 I , for

I1 V '2'''lk(X)AI 1'2''''k)

converges normally on the compact subsets of the open set A which contains the origin, then it represents a left O-anallytic function I in A. Moreover the given series is precisely the Taylor series about the origin of Notice that

I

in A (It means that AI11 2 ••• i,

= 0 ¢:::> DI(x -

DI(x)

a)

= OX'l

= 0,

.•. OX'k 1(0)).

\:Ix E 0,

we have the general Taylor type theorem: Theorem 3 If I is left O-analytic in an open set n, then for each point a E n there exists a neighbourhood Au C n, in which I may be developed into a unique normally convergent Taylor series of left spherical O-analytic functions 00

I(x) Pk(x)

=L =

o 1 k!

PkI(x),

L 7

11,1 2 ,

L

••• ,lk=O

(Xlk -a' k)'''(X' 1 -a,J ox'l ",ox,J(a)

Li & Peng: TAYLOR SERIES AND ORTHOGONALITY OF FUNCTIONS

No.3

z/(a)

= (x/ -

329

a/leo - (xo - ao)e" 1 = 1,2"",7.

And we also have Theorem 4 Let n be an open connected set in R 8 , and the following statements are equivalent:

f

be left O-analytic in n. Then

1) f(x) =: 0, xEn; 2) There exists a point a E n such that

4

The Orthogonality of O-Analytic FUnctions Let M be an 8-dimensional, compact, oriented

in some open connected subset n of R

8

For j :

.

Coo~manifold with

°:S j :S 7, Let

boundary oM contained 7

dij = dxo !\ dXl !\ ... dXj_l !\ dXj+l !\ ... !\ dX7, Theorem 5

r

JaM

dcr(x) = ~) -l)j ejdij o

If t, g E C1(n, 0), then

f(dcrg) + (gdcr)f =

r [(lD)g + (gD)f +.f(Dg) + g(DJ)]dV.

JM

Proof By Stokes' theorem

Where dV(x) = dxo !\ ... !\ dX7 is the volume elements on n. Similarly, we have

1

8M

(gdcr)f

=

1 M

(gD)fdV

+

.

L M

g(Df)dV

+

IL M

i,A,B

leA, ei, eBl) ~~ gBdV. I

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ACTA MATHEMATIC A SCIENTIA

Thus

f

JoM

!(do-g)

+ (gdo-)! = f

JM

[(fD)g

Vo1.21 Ser.B

+ (gD)! + !(Dg) + g(Df)]dV.

If D! =!D = gD = Dg = 0, then JoM !(do-g) + (gdo-)! = o. Corollary 2 If D! ! D 0, then JoM !do- ! = O. and V"'2''''h are both left and right O-analytic functions, we have Since V,"2""h Corollary 3 Let V(h12" ·lk) stand for V,"2""h or V'l'2···'h. Then Corollary 1

f

JoM

=

=

[V(1112'" h)(do-V(sls2'" st})

+ (V(SlS2'"

st}do-)V(h12" ·lk)]dV

=0

We can prove that if Dg = 0 and! = XkeO - XOek, k = 1,2,···7 or ! (XkeO - xOek)n, (n E N) or ! = (XkeO - xOek)(x,eo - xoe,), (k:f I) then Remark

f

JoM

!(do-g)

= O.

We conjecture that: if Dg = 0, then JoM V(1}12" ·lk)(do-g) conjecture is that: if g = V(SlS2'" St), then

= O.

The weak version of the

References 1 Jacobson N. Basic Algebras 1. Second Edition. New York: W H Freeman and Company, 1985

2 Futer R. Die Funktionentheorie der Der Differential gleichungen Au = 0 und AAu = 0 mit vierrealen Variablen. Conun Math Helv , 1934, 7: 30117-308 3 Futer R. Zur Theotie der regulaaren Funktioneneiner Quaternionenvariablem. Monat fiir Math und Phys, 1935, 43: 69-74 4 Clifford W K, Application of Grassman's extensive algebra. Amer J of Math, 1878, 1: 350-358 5 Brackx F, Dlanghe R, Sonunen F. Clifford Analysis. Pitman Advanced Publishing Program, 1982 6 Mitrea M. Clifford Wavelets, Singular integrals, and Hardy spaces. Lecture Notes in Math, 1575. SpringerVerlag, 1994 7 Gilbert J E, Murray M A M. Clifford algebras and Dirac operators in harmonic analysis. Cambridge: Cambridge University press, 1991 8 Hestenes D, Sobczyk D. Clifford Algebral to Geometric Calculus, A unified language for mathematics and physics. Reidel D, Pull, Dordrecht, Boston, Lancaster, 1984 9 Delanghe R, Sonunen F, Soucek V. Clifford Algebra and Spinor-Valued Functions, a function theoty for the Dirac Operater. London:Kluwer Academic Publishers, 1992 10 Diindarer D, Giirsey F, Tze C H. Self-duality and Octonionic analyticity of 8 7 -valued antisymmetric fields in eight dimensions. Nuclear Physics, 1986, 266B: 440-450 11 Li Xingmin, Peng Lizhong. The Cauchy integral formulas on the Octonions. Bull Soc Math Belg (to appear) 12 Li Xingmin, Peng Lizhong. The 3-line theorems on the Octonions. Advance in Math, 2000, 30: 83-85 13 Li Xingmin. On the two questions in Clifford analysis and Octonion analysis. Finite or Infinite Dimensional Complex Analysis: 7th International Colloquium. Dekker, NY, 2000: 277-285

Lecture Notes in Pure and Applied Series, Marcel

14 Li Xingmin, Peng Lizhong. On Stein-Weiss conjugate harmonic function and Oct onion analytic function. Approx Theory and its Appl, 2000, 16: 28-36 15 Peng Lizhong, Yang Lei. The curl in 7-dimensional space and its applications. Approx Theory and its Appl, 1999, 15: 66-80