Multidimensional Laplace Transforms over Quaterni o ns , Octonions and Cayley-Dickson Algebras , Their Applications to PDE

Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.


Introduction
The Laplace transform over the complex field is already classical and plays very important role in mathematics including complex analysis and differential equations [1][2][3].The classical Laplace transform is used frequently for ordinary differential equations and also for partial differential equations sufficiently simple to be resolved, for example, of two variables.But it meets substantial difficulties or does not work for general partial differential equations even with constant coefficients especially for that of hyperbolic type.
To overcome these drawbacks of the classical Laplace transform in the present paper more general noncommutative multiparameter transforms over Cayley-Dickson algebras are investigated.In the preceding paper a noncommutative analog of the classical Laplace transform over the Cayley-Dickson algebras was defined and investigated [4].This paper is devoted to its generalizations for several real parameters and also variables in the Cayley-Dickson algebras.For this the preceding results of the author on holomorphic, that is (super) differentiable functions, and meromorphic functions of the Cayley-Dickson numbers are used [5,6].The super-differentiability of functions of Cayley-Dickson variables is stronger than the Fréchet's differentiability.In those works also a noncommutative line integration was investigated.
We remind that quaternions and operations over them had been first defined and investigated by W. R. Hamilton in 1843 [7].Several years later on Cayley and Dickson had introduced generalizations of quaternions known now as the Cayley-Dickson algebras [8][9][10][11].These algebras, especially quaternions and octonions, have found applications in physics.They were used by Maxwell, Yang and Mills while derivation of their equations, which they then have rewritten in the real form because of the insufficient development of mathematical analysis over such algebras in their time [12][13][14].This is important, because noncommutative gauge fields are widely used in theoretical physics [15].
Each Cayley-Dickson algebra r A over the real field R has 2 r generators   A  is formed from the preceding algebra r A with the help of the so-called doubling procedure by generator 2 r i .In particular, 1 = A C coincides with the field of complex numbers, 2 = A H is the skew field of quaternions, 3 A is the algebra of octonions, 4 A is the algebra of sedenions.This means that a sequence of embeddings The multiplication in 1 r A  is defined by the following equation: . At the beginning of this article a multiparameter noncommutative transform is defined.Then new types of the direct and inverse noncommutative multiparameter transforms over the general Cayley-Dickson algebras are investigated, particularly, also over the quaternion skew field and the algebra of octonions.The transforms are considered in r A spherical and r A Cartesian coordinates.At the same time specific features of the noncommutative multiparameter transforms are elucidated, for example, related with the fact that in the Cayley-Dickson algebra r A there are 2 1 r  imaginary generators , , r i i   apart from one in the field of complex numbers such that the imaginary space in r A has the dimension 2 1  r  .Theorems about properties of images and originals in conjunction with the operations of linear combinations, differentiation, integration, shift and homothety are proved.An extension of the noncommutative multiparameter transforms for generalized functions is given.Formulas for noncommutative transforms of products and convolutions of functions are deduced.
Thus this solves the problem of non-commutative mathematical analysis to develop the multiparameter Laplace transform over the Cayley-Dickson algebras.Moreover, an application of the noncommutative integral transforms for solutions of partial differential equations is described.It can serve as an effective means (tool) to solve partial differential equations with real or complex coefficients with or without boundary conditions and their systems of different types (see also [16]).An algorithm is described which permits to write fundamental solutions and functions of Green's type.A moving boundary problem and partial differential equations with discontinuous coefficients are also studied with the use of the noncommutative transform.
Frequently, references within the same subsection are given without number of the subsection, apart from references when subsection are different.
All results of this paper are obtained for the first time.

Definitions Transforms in A r Cartesian Coordinates
Denote by r A the Cayley-Dickson algebra, 0 r  , which may be, in particular, we call a function-original, where 2 r  , n N  , if it fulfills the following conditions (1-5).
1) The function   , , j n  , everywhere on n R may be besides points of discontinuity of the first type.
4) The function   f t increases not faster, than the exponential function, that is there exist constants , a a R   , where , where   is the standard basis of generators of r A so that 0 = 1 i , 2 = 1 is the parameter of an initial phase, j R   for each = 0,1, , 2 1 we also put for the general Cayley-Dickson algebra with  u p t  is given by Formulas (1,2).
At the same time the components j p of the number p and j  for  in   , ; u p t  we write in the pand  -representations respectively such that 5) Henceforth, the functions   , ; u p t  given by 1 (8,8.1)or (1,2,2.1)are used, if another form (3) is not specified.

If for  
, ; u p t  concrete formulas are not mentioned, it will be undermined, that the function   , ; u p t  is given in r A spherical coordinates by Expressions 1,2,2.1).If in Formulas 1 (7) or (4) the integral is not by all, but only by ( , then we denote a noncommutative transform by ; , ; ;  if something other is not specified.

Remark
The spherical r A coordinates appear naturally from the following consideration of iterated exponents: Consider 2 r i the generator of the doubling procedure of the Cayley-Dickson algebra M p t  from Definition 2 over r A in more details by r M .Then by induction we write:   Formulas 1 (7) and 2(4) define the right multiparameter transform.Symmetrically is defined a left multiparameter transform.They are related by conjugation and up to a sign of basic generators.For real valued originals they certainly coincide.Henceforward, only the right multiparameter transform is investigated.
Proof.At first consider the characteristic functions

Corollary
Let suppositions of Theorem 4 be satisfied.Then the image

 
, ; ; u u p t  given by 2 (1,2) has the following periodicity properties:   for each s j  and 1 s j   , while either for each n t R  .On the other hand, either , where . From this and Formulas 2 (1,2,4) the second and the third statements of this corollary follow.

Remark
For a subset U in r A we put is the family of standard ge-

 
, := : , Recall that in § § 2.5-7 [6] for each continuous function : it was defined the operator f by each variable r z A  .For the non-commutative integral transformations consider, for example, the left algorithm of calculations of integrals.
A Hausdorff topological space X is said to be nconnected for 0 n  if each continuous map : k f S X  from the k -dimensional real unit sphere into X has a continuous extension over  (see also [17]).A 1-connected space is also said to be simply connected.
It is supposed further, that a domain U in r A has the property that U is   , for which there exists = z u t U   .Then at each point t, where   f t satisfies the Hölder condition the equality is accomplished : and 2), the integrals are taken along the straight lines

Theorem
< = < a Re p a a  and this integral is understood in the sense of the principal value, Proof.In Integral (1) an integrand   p dp  certainly corresponds to the iterated integral as , where . Using Decomposition 3(3.1) of a function f it is sufficient to consider the inverse transformation of the real valued function j f , which we denote for simplicity by f .We put

 
Int U of the domain U there is accomplished the equality , where the inte- gral depends only on an initial 0 z and a final z points of a rectifiable path in   0 , , r B A z R , a R  (see also Theorem 2.14 [4]).Therefore, along the straight line j N R the restriction of the antiderivative has the form Using Formula 4(1) we reduce the consideration to for each real numbers , , ,     and a purely imaginary Cayley-Dickson number M .The octonion algebra O is alternative, while the real field R is the center of the Cayley-Dickson algebra r A .We consider the integral 3) for each positive value of the parameter 0 < < b  .With the help of generators of the Cayley-Dickson algebra r A and the Fubini Theorem for real valued components of the function the integral can be written in the form: A (see also Proposition 2.18 [4]).If take marked k t for each k j  and = j S N for some 1 j  in Lemma 2.17 [4] considering the variable j t , then with a suitable ( R -linear) automorphism v of the Cayley-Dickson algebra r A an expression for  

 
, ; v M p t  simplifies like in the complex case with : The latter identity can be applied to either     . We take the limit of   b g t when b tends to the infinity.Evidently, , , , ; for N u given by 2(1,2,2.1),where j M is prescribed by (7),

 
, , = ; u p   given by Formulas 2(1,2,2.1) or u p   described in 1(8,8.1).Then the integral operator (see also Formula (4) above) applied to the function , , ; with the parameter j  instead of  treated by Theo- rems 2.19 and 3.15 [4] gives the inversion formula corresponding to the real variable j t for   f t and to the Cayley-Dickson variable 0 0 with the help of Formulas (6-10) and 3(1,2) we get the following: Re f f for each q and in (11) the function = q f f stands for some marked q in accordance with Decompositions 3(3,3.1)and the beginning of this proof.
Mention, that the algebra alg N N N over the real field with three generators . On the other hand, . We use decompositions (7-10) and take 2 = k l due to Formula (11), where Re stands on the right side of the equality, since for each k l  .Thus the repeated application of this procedure by = 1, 2, , j n  leads to Formula (1) of this theorem.

Corollary
If the conditions of Theorem 6 are satisfied, then .
alg N N N is alternative.Therefore, in accordance with § 6 and Formulas 1(8,8.1)and 2(1-4) for each non-commutative integral given by the left algorithm we get 2) for each = 1, , j n  , since the real field is the center of the Cayley-Dickson algebra r A , while the functions sin and cos are analytic with real expansion coefficients.Thus 3) hence taking the limit with b tending to the infinity implies, that the non-commutative iterated (multiple) integral in Formula 6(1) reduces to the principal value of the usual integral by real variables   is the image of the function, 4) , , ; . .An integration by dp in the iterated integral ( 4) is treated as in § 6.Take marked values of variables 1 u p t  given by 1(8,8.1)instead of  and any non-zero Cayley-Dickson number For any locally z-analytic function   g z in a domain U satisfying conditions of § 5 the homotopy theo-rem for a non-commutative line integral over r A , 2 r  , is satisfied (see [5,6]).In particular if U contains the straight line j w RN  and the path  is a finite number (see Lemma 2.23 in [4]).We apply this to the integrand in Formula (4), since is locally analytic by p in accordance with Theorem 4 and Conditions (1,2) are satisfied.
Then the integral operator    on the j -th step with the help of Theorems 2.22 and 3.16 [4] gives the inversion formula corresponding to the real parameter j t for   as in § 6 implies Formula (4) of this theorem.Thus there exist originals 0 f and 1 f for functions F p  due to the distributivity of the multiplication in the Cayley-Dickson algebra r A leading to the additivity of the considered integral operator in Formula (4) .

Corollary
Let the conditions of Theorem 8 be satisfied, then , , ; .
In accordance with § § 6 and 6.1 each noncommutative integral given by the left algorithm reduces to the principal value of the usual integral by the corresponding real variable: where . Indeed, this condition leads to the accomplishment of the r A analog of the Jordan Lemma for each 2 r  (see also Lemma 2.23 and Remark 2.24 [4]).
Subsequent properties of quaternion, octonion and general r A multiparameter non-commutative analogs of the Laplace transform are considered below.We denote by: 2) Proof.Since the transforms We have r r n     , while R is the center of the Cayley-Dickson algebra r A .The quaternion skew field H is associative.Thus, under the imposed conditions the constants ,   can be carried out outside integrals.

Theorem
for each = 1, , j n  .Then changing of these variables implies: due to the fact that the real filed R is the center Z(A r ) of the Cayley-Dickson algebra r A .

Theorem
Let   f t be a function-original on the domain , ; ; = , , ; ; ; , ; ; = , , ; ; ; , ; ; , where for each Formulas 30(6,7) [4] we have the equality in the r A spherical coordinates: j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j e j j j j j j j j p s p s i s p s i s s and k s are real independent variables for each k j  , where , = 0 cos sin e j j j j j j j j j j j j j j j j

 
= e e j j j S S  , the zero power 0 = e j S I is the unit operator; in the r A spherical coordinates, where either = 1 q or = 1 q  and 3.6) and any real number x R  , where 1 j  .Then in accordance with Formula (3.2) we have: u p t  given by Formulas 1(8,8.1) in the r A Cartesian coordinates, where either = 1 q or = 1 q  .The integration by parts theorem (Theorem 2 in § II.2.6 on p. 228 [18]) states: if < a b and two functions f and g are Riemann integrable on the segment   , where A and B are two real constants, then Therefore, the integration by parts gives 4) , , , , and applying the Fubini's theorem componentwise to j j f i we infer: for each 1 j n   .This gives Formula (1), where 6) 1, ,1 0 0 , , ; ; ; = exp , ; d = , , ,0, , ,  in real variables are also frequently used in the class of infinite differentiable functions with converging Taylor series expansion in the corresponding domain.
It is possible to use also the following convention.One can put gives with such convention the same result as can be used.

n n n u p t p h u p t s h S u p t s h S u p t s h
in the r A Cartesian coordinates.Thus from Formulas (2,3) we deduce Formula (1).

Theorem
, ; ; = d , where = t    .Symmetrically we get (2) for v U instead of 1, ,1 U  .Naturally, that the multiparameter non-commutative Laplace integral for an original f can be considered as the sum of 2 n in-tegrals by the sub-domains v U : The summation by all possible gives Formula (1).

Note
In view of the definition of the non-commutative transform n F and   , ; u p t  and Theorem 14 the term has the natural interpretation as the initial phase of a retardation.

Theorem
Let a function   f t be a real valued original, u p t  is given by 1(8,8.1)or 2(1,2,2.1).Let also

 
; G p  and   q p be locally analytic functions such that 1) , then in view of the Fubini's theorem and the theorem conditions a change of an integration order gives the equalities: and the center of the algebra r A is R .

Theorem
given by 2(1,2,2.1),then u p t  given by 1(8,8.1),where , p tends to the infinity inside the angle in the r A Cartesian coordinates, where 0 p  inside the same angle.
We apply these Formulas (3,4) which gives the first statement of this theorem, since for each 1 j n   .Therefore, the limit exists: from which the second statement of this theorem follows in the r A spherical coordinates and analogously in the r A Cartesian coordinates using Formula (3.1).

Definitions
Let X and Y be two R linear normed spaces which are also left and right r A modules, where 1 r  .Let Y be complete relative to its norm.We put : we denote a family of all continuous k times R poly-linear and A r additive operators from  is also a normed R linear and left and right r A module complete relative to its norm.In particular,  , then an operator A we call right or left r A -linear respectively.An R linear space of left (or right) k times r A poly-linear operators is denoted by We consider a space of test function . , , = ., , and every transposition .
, , = , where   = 0, , 0,1, 0, , 0 The set of all such functionals is denoted by converges to zero for n tending to the infinity.
A generalized function g is zero on an open subset

 
t supp g  the functional g is different from zero.The addition of generalized functions , g h is given by the formula: The multiplication ' g D  on an infinite differentiable function w is given by the equality: A generalized function g prescribed by the equation: Another space denotes a ball with center at z of radius R in a metric space Z with a metric  .The family of all R-linear and r A -additive functionals on B is denoted by B .
In particular we can take . By an image of such original we call a function.5) by the following rule.For a given where in each term  by Condition (4), while the sum in ( 6) is by all admissible vectors

Note and Examples
Evidently the transform

 
, ; ; , , , , , , At the same time the real field R is the center of the Cayley-Dickson algebra r A , where 2 r N   .Let  be the Dirac delta function, defined by the equation since it is possible to take In the general case: 3)   f t can not be transformed.

Theorem
given by 2(1,2, 2.1), or , ; ; = , ; ; < < a Re p a  the first additive is zero, while the second integral converts with the help of Formulas 12(2,2.1),Formula (1) follows for = 1 k : 3) , ; ;  , ; ; To accomplish the derivation we use Theorem 14 so that  < < a Re p a  , when it is non void, Formula (3) is valid.Applying Formula (3) in the r A spherical coordinates by induction to with the corresponding order subordinated to , or in the r A Cartesian coordinates using Formula 12(1.1) for the partial derivatives ) : ,0 with the corresponding order subordinated to we deduce Expressions ( 1) and (1.1) with the help of Statement 6 from § XVII.2.3 [19] about the differentiation of an improper integral by a parameter and § 2.

Remarks
For the entire Euclidean space n R Theorem 21 for   , ; ; For it evidently Theorems  = ,

 
= ; j j s s n t as in § 2, or applying to the partial derivatives = 0 lim In accordance with Note 8 [4]  A Cartesian coordinates, which gives the statement of this theorem.

Theorem
, , for each k , where Let the vector   l enumerate faces ( ) (see also more detailed notations in § 28).

 
= e e j j R R p are operators depending on the parameter p .If ( ) = l j t  for some 1 j n   , then the corresponding addendum on the right of (1) is zero.
Proof.In view of Theorem 23 we get the equality 2) On the other hand, for p W  additives on the right of (2) convert with the help of Formula 23(1).Each term of the form can be further transformed with the help of (2) by the considered variable k t only in the case = 0 k l .Applying Formula (2) by induction to partial derivatives as in § 21 and using Theorem 14 and Remarks 22 we deduce (1).

 
, ; ; of an original function

 
f t for u given by 2(1,2,2.1) in the half space , where 1 a is fixed, , ; ; , in the A r Cartesian coordinates correspondingly for each 1 j  .Proof.Take a path of an integration belonging to the half space  is absolutely converging and the limit uniformly by z on each compact subset in r A , where S is a purely imaginary marked Cayley-Dickson number with = 1 S .Therefore, in the integral 6) the order of the integration can be changed in accordance with the Fubini's theorem applied componentwise to an integrand  in the r A Cartesian coordinates for each 2 j  is essential for the convergence of such integral.We certainly have 8) for each > 0 in the r A Cartesian coordinates.

Application of the Noncommutative Multiparameter Transform to Partial Differential Equations
Consider a partial differential equation of the form: are continuous functions, where 0 , the operator A can be rewritten in s coordinates as Particularly, the entire space n R may also be taken.
Under the linear mapping     , , , , We consider a manifold W satisfying the following conditions (i-v).
i).The manifold W is continuous and piecewise C  , where l C denotes the family of l times continuously differentiable functions.This means by the definition that W as the manifold is of class 0 . For > 0 j the set j W is allowed to be void or non-void.iv).A sequence   , exists such that k W uniformly converges to W on each compact subset in n R relative to the metric dist .
For two subsets B and E in a metric space X with a metric  we put


. We shall consider surface integrals of the second kind, i.e. by the oriented surface W (see (iv)), where each j W , = 0, , where are given functions.Generally these conditions may be excessive, so one uses some of them or their linear combinations (see (5.1) below).Frequently, the boundary conditions U is a ball in n R with the center at zero, can also be specified, if the boundary conditions demand it.
The complex field has the natural realization by 2 2  real matrices so that 0 . The quaternion skew field, as it is well-known, can be realized with the help of 2 2  complex matrices with the gene- , ; ; = , ; ; , ; ; in accordance with § 25 and the notation of this section.
Therefore, Equation (6) shows that the boundary conditions are necessary: , , m t t We choose them so that each two different quadrants may intersect only by their borders, each k U satisfies the same conditions as U and 7) Therefore, Equation ( 6) can be written for more general domain U also.
For U instead of n Q we get a face Thus the sufficient boundary conditions are: are necessary for = < q   and q as above.Depending on coefficients of the operator A and the domain U some boundary conditions may be dropped, when the corresponding terms vanish in Formula (6).For example, if is not necessary, only the boundary condition  , ; ; , ; ; ; , ; ; : and polynomials of p , where Z denotes the ring of integer numbers, where the corresponding term  for some j .In the r A Carte-sian coordinates there are not so well periodicity properties generally, so the family may be infinite.This means that     , ; ; n F f t u p  can be expressed in the form:

 
,( ),( ),( ) j q l m P p are quotients of polynomials of real variables 0 1 , , , n p p p  .The sum in ( 8) is finite in the r A spherical coordinates and may be infinite in the A r Cartesian coordinates.To the obtained Equation ( 8) we apply the theorem about the inversion of the noncommutative multiparameter transform.Thus this gives an expression of f through g as a particular solution of the problem given by (1,2,3.1)and it is prescribed by Formulas 6.1(1) and 8.1 (1).For

 
; ; ; ,( ),( ),( ) j q l m P p are quotients of polynomials with real, complex or quaternion coefficients and real variables, also n G and terms on the right of ( 6) satisfy them.Thus we have demonstrated the theorem.

 
; ; ; Mention, that a general solution of (1,2) is the sum of its particular solution and a general solution of the Af g and j f on

Example
We take the partial differential operator of the second order where the quadratic form    is nondegenerate and is not always negative, because otherwise we can consider A  .Suppose that = , where     , ; ; = , ; ; , ; ; , ; ; = , ; ; in the r A spherical or r A Cartesian coordinates, where  with 1 on the h -th place, 0 = S I is the unit operator, the operators R p are given by Formulas 25(1.1) or 25 (12) respectively.
We denote by where  denotes a volume element on the m dimensional surface S (see Condition (v) above).Thus we can consider a non-commutative multiparameter transform on U  for an original f on U given by the formula: 11) , ; ; (10) correspond to the boundary n Q  .They can be simplified: ;  on the corresponding faces of n Q orthogonal to n e given by condition: either , then this induces the corresponding embedding  of n Q or U into r A .This permits to make further simplification: , so that

 
M t is a purely imaginary Cayley-Dickson number,

 
a t is a piecewise constant function equal to h a for the corresponding t in the face  is an original on U whenever this in-tegral exists.For example, when  is a linear combination of shift operators ( ) .
Using Conditions (iv-vii) and the sequence m U and quadrants , n m l Q outlined above we get for a boundary problem on U instead of n Q the following equation: , ; ; = , ; ; , ; ; = , ; ; , (see also Stokes' formula in § XIII.3.4 [19] and Formulas (14.2,14.3)below).Particularly, for the quadrant domain and zero otherwise.The boundary conditions are: The functions   a t and   , where   and its external product with This is sufficient for the calculation of

Inversion Procedure in the A r Spherical
Coordinates When boundary conditions 28(3.1) are specified, this Equation 28( 6) can be resolved relative to , particularly, for Equa- tions 28.2(14,14.1)also.The operators e j S and j T of § 12 have the periodicity properties: ; = ; .
Then from Formula 28(6) we get the following equations: These equations are resolved for each = 1, , w n  as it is indicated below.Taking the sum one gets the result 6.4) The analogous procedure is for Equation ( 14) with the domain U instead of n Q .From Equation (6.3) or ( 14) we get the linear equation: 15 where  is the known function and depends on the parameter  , ( )  are known coefficients depending on p , ( ) x are indeterminates and may depend on  , = Acting on both sides of (6.3) or ( 14) with the shift operators ( ) m T (see Formula 25(SO)), where 1 = 0,1 linear equations with the known functions Each such shift of  left coefficients ( )  otherwise.This system can be reduced, when a minimal additive group generated by all with non-zero coefficients in Equation ( 15) , where

 
:= h g Z hZ denotes the finite additive group for 0 < h Z  .Generally the obtained system is non-degenerate for or in W , where 1 n   denotes the Lebesgue measure on the real space We consider the non-degenerate operator A with real, complex i C or quaternion , , J K L H coefficients.Certainly in the real and complex cases at each point p , where its determinate is non-zero, a solution can be found by the Cramer's rule.Generally, the system can be solved by the following algorithm.We can group variables by 1 2 , , , k l l l  .For a given 2 , , h l l  and 1 = 0,1 l subtracting all other terms from both sides of (15) after an action of ( ) m T with 1 = 0,1 m and marked h m for each > 1 h we get the system of the form 16) which generally has a unique solution for , , , it can be solved by the Gauss' exclusion algorithm.In the first two cases of R or i C the solution is: x   , where Thus on each step either two or four indeterminates are calculated and substituted into the initial linear algebraic system that gives new linear algebraic system with a number of indeterminates less on two or four respectively.May be pairwise resolution on each step is simpler, because the denominator of the type    6), ( 14) above).This algorithm acts analogously to the Gauss' algorithm.Finally the last two or four indeterminates remain and they are found with the help of Formulas either (17) or (19) respectively.When for a marked h in ( 6) or (14) of convergence of the noncommutative multiparameter transform, when it is non-void, . This domain W is caused by properties of g and initial conditions on U  and by the domain U also.Generally U is worthwhile to choose with its interior

 
Int U non-intersecting with a characteristic surface and at least one of the partial derivatives   0 In particular, the boundary problem may be with the right side 14), where  is a real or multiplier, when boundary conditions are non-trivial.In the space either

Examples
Take partial differential equations of the fourth order.In this subsection the noncommutative multiparameter transforms in r A spherical coordinates are considered.For 20) B R A , where 2 n  , Equation ( 6) takes the form: , ; ; = 3 , ; ; , ; ; = , ; ; due to Corollary 4.1.In accordance with (16,17) we get: ; From Theorem 6, Corollary 6.1 and Remarks 24 we infer that:   of the type described by ( 16) with so-lutions given by Formulas (17).It is seen that these coefficients are non-zero Finally Formula (23) provides the expression for f on the corresponding domain W for suitable known function g for which integrals converge.If For (21,24) on a bounded domain with given boundary conditions equations will be of an analogous type with a term on the right     , ; ; n F g t u p  minus boundary terms appearing in (6) in these particular cases.
In the particular case, when for each 1 n t  , p , t and  , with the help of (6,8) one can deduce an expression of ; ; := , , , ; ; and boundary terms in the following form: since the octonion algebra is alternative and each equation = bx c with non-zero b has the unique solution C is an octonion constant which can be specified by an initial condition.More general partial differential equations as (30), but with of the right side of (33) one gets the particular solution f .

 
, ; ; the phase shift operator is isometrical: 43) (see § 12).In the r A Cartesian coordinates each Cayley-Dickson number can be presented as: 42.1) , where R   is a real parameter, M is a purely imaginary Cayley-Dickson number (see also § 3 in [5,6]).Therefore, we deduce that 44) Then expressing  from (40) and using Formulas (41, 42, 42.1, 44) we infer, that 45)   , , ; For a calculation of the appearing integrals the generalized Jordan lemma (see § § 23 and 24 in [4]) and residues of functions at poles corresponding to zeros

The Decomposition Theorem of Partial Differential Operators over the Cayley-Dickson Algebras
We consider a partial differential operator of order u : 1) Speaking about locally constant or locally differentiable coefficients we shall undermine that a domain U is the union of subdomains j U satisfying conditions 28(D1,i-vii) and = . Therefore, it remains the case of the operator A of the even order = 2 u s.Take , ,, 0 , ,  

Corollary 1
Let suppositions of Theorem 29 be satisfied.Then a change of variables locally constant or variable 1  C or x-differentiable on U correspondingly exists so that the principal part A is presented in accordance with Formulas 29(4,5), then three operators s  ,

Products of Operators
We consider operators of the form:

Fundamental Solutions
Let either Y be a real son algebra (see § 28).Consider the space   , n B R Y (see § 19) supplied with a topology in it is given by the countable family of semi-norms 1) We define their convolution as x y R  due to (4), since the latter Equality ( 5) is satisfied for each pair j f and k g .Thus a solution of the equation , where denotes a fundamental solution of the equation The fundamental solution of the equation 9 using Equalities 32(2-4) can be written as the convolution 10) More generally we can consider the equation 11 , ,    are operators of orders s, 1 s and 2 s respectively given by 32(1) with z-differentiable coefficients.For 2 2 = 0    this equation was solved above.Suppose now, that the operator 2 2    is non-zero.To solve Equation (11) on a domain U one can write it as the system: Find at first a fundamental solution A V of Equation ( 11) for = g  .We have: 13) In accordance with (3-5) and 32(1) the identity is satisfied: Thus ( 13) is equivalent to . We consider the Fourier transform F by real variables with the generator i commuting with j i for each = 0, , 2 1 x R  for every j .The inverse Fourier transform is: For a generalized function f from the space In view of (2-5) the Fourier transform of ( 14) gives:   the latter equation gives the linear system of g can be found, since    (see also the Fourier transform of real and complex generalized func-tions in [1,21]).Then 16)

and = *
A f V g gives the solution of (11), where 1 g was calculated from (15).Let be the R -linear projection operator defined as the sum of projection operators , where   π : gives the corresponding restrictions when A on the final step are with constant coefficients.A residue term Q of the first order can be integrated along a path using a non-commutative line integration over the Cayley-Dickson algebra [5,6].

Examples
, by the variables   ; .
for any real number.
The functions ; ; ; for each k .Particularly, we take = 1 Thus the inverse Laplace transform for 0 = 0 q and = 0  in accordance with Formulas 2(1-4) reduces to in the class of the generalized functions is known (see [21] and § § 9.7 and 11.8 [1]) and gives   and any 1 j n   in accordance with the conditions imposed on j c at the beginning of this section and = j j iN N i for each j , the Fourier transform with the generator i can be accomplished subsequently by each variable using Identity (19) .We treat the iterated integral as in § 6, i.e. with the same order of brackets.Taking initially j c R  and considering the complex analytic extension of formulas given above in each complex plane j R N R  by j c for each j by induction from 1 to n , when j c is not real in the operator A ,   given by Formulas 2 (5,6) and 33 (17).We define the following operators  Proof.The product of two originals can be written in the form: p s p q s i p q s q s i q s j j j j j j j j j j j j j j j j j n n n n R R R R f t g t e t f t e t g t e t q .We get the integration by 1 , , n q q  , which gives convolutions by the  f t w y t y t t (See reference [21][22][23][24][25][26][27][28][29][30]).

Partial Differential Equations with Discontinuous Coefficients
Consider a domain U and its subdomains   and are allowed to be discontinuous at the common borders . Thus the equivalent problem is: 3 Cayley-Dickson algebras have a natural physical meaning as generating operators of fermions.The skew field of quaternions is associative, and the algebra of octonions is alternative.The Cayley-Dickson algebra r A is power associative, that is, = .It is nonassociative and non-alternative for each 4 r  .A conjugation * = z z  of Cayley-Dickson numbers The octonion algebra has the multiplicative norm and is the division algebra.Cayley-Dickson algebras r A with 4 r  are not division algebras and have not multiplicative norms.The conjugate of any Cayley-Dickson number z is given by the formula:

2 = H A the quaternion skew field or 3 =
O A the octonion algebra.For unification of the notation we put 0 = A R

F
p is called the noncommutative multiparameter (Laplace) transform at a point r p A  of the function-original   the more general non-commutative multiparameter transform over r A is defined by the formula: 4) as the principal value of either Riemann or Lebesgue integral, the image of the function-original j f .If an automorphism of the Cayley-Dickson algebra r A is taken and instead of the standard generators new basic generators, where 2 r N   .In this more general case we denote by   ; n N u F p  an image for an original   f t , or in more details we denote it by from the 2π periodicity of sine and cosine functions the first statement follows.From its image multiparameter noncommutative transform, where the functions f and n u F are written in the forms given by 3(3,3.1), a holomorphic function of the Cayley-Dickson variable, then locally in a simply connected domain U in each ball   For the chosen branch of the line integral specified by the left algorithm this antiderivative is unique up to a constant from r A with the given z -representation  of the function  [4-6].On the other hand, for analytic functions with real expansion coefficients in their power series non-commutative integrals specified by left or right algorithms along straight lines coincide with usual Riemann integrals by the corresponding variables.The functions the multiparameter non-commutative transform are analytic with real expansion coefficients in their series by powers of r z A  .
j N , k N and l N is alternative.The product k l N N of two generators is also the corresponding generator   l and the sign multiplier misunderstanding we shall use e j S and j T in the sense of Formulas 12(3.1-3.7).It is worth to mention that instead of 12(3.7) also the formulas 1) the j-th place.Such convergence in D defines closed subsets in this space D , their complements by the definition are open, that gives the topology on D .The space D is R linear and right and left r A module.By a generalized function of class

1 =
since the right side of Equation 19(5) is holomorphic by p in f W and by  in view of Theorem 4. Equation 19(5) implies, that Theorems 11-13 are accomplished also for generalized functions.For 1 a a  the region of convergence reduces to the vertical hyperplane in r A over R .common domain of convergence and  

b
Re p  for = k b  for some k and finite k a for each k ;  for some k and l ;  

R
are given by Formulas 25(1,1,1.2).Proof.In view of Theorem 25 the equation 3) Formulas (3-9) and 2(1,2,2.1)or 1(8,8.1)and 12(3.1-3.7)we deduce that: identically on the corresponding domain V .We consider that (D1) U is a canonical closed subset in the Euclidean space n R , that is     = U cl Int U , where   Int U denotes the interior of U and   cl U denotes the closure of U.
in the r A spherical or r A Cartesian coordinates, where the operators   j e R p are given by Formulas 25(1.1) or 25(1.2).Here   l enumerates faces ( )

1 Equation ( 6 )
then no any boundary condition appears.are along orthogonal to them coordinates in n R , so they are correctly posed.In r A spherical coordinates due to Corollary 4.with different values of the parameter gives a system of linear equations relative to unknown functions

nF f u p  given by the right side of ( 8 )
satisfies Conditions 8(3).Then Problem (1,2,3.1)has a solution in the class of original functions, when g and ,( ) l   are originals, or in the class of generalized functions, when g and ,( ) l   are generalized functions.

3 A
 .Then we reduce this form   a  by an invertible R linear operator C to the sum of squares.
originals or f and g are generalized functions.For two quadrants , common face  external normals to it for these quadrants have opposite directions.Thus the corresponding integrals in 1 ,

=
cx bx ax b    with coefficients a , b and c , and Cayley-Dickson numbers on the right side R is the center of the Cayley-Dickson algebra Formulas 12(3.1-3.7).
denotes the space of k times continuously differentiable functions by all real variables 0 , , n x x  on U with values in r A , while the x -differentiability corresponds to the super-differentiability by the Cayley-Dickson variable x .

1 C
 transforms the right side of Formula (4), when it is written in the Therefore, we deduce in accordance with (5) and 2(3,4) and Corollary 6.1 with parameters 0 = 0 p Cartesian coordinates, since for any even function its cosine Fourier transform coincides with the Fourier transform.The inverse Fourier transform         of § 2 over the Cayley-Dickson algebra has permitted to get the solution of the Laplace operator.3 in [21]).The function P  has the cone surface   zeros, so that for the correct definition of generalized functions corresponding to P  3 n  (see § IV.2.6 [21]), where   , = det j k D g denotes a discriminant of the quadratic form   number can be presented in the polar form= M z z e  , R   , π   , M is a purely imaginary Cayley-Dickson number = 1 M, the countable number of values, k Z  (see § 3 in[5,6]).Therefore, we choose the branch the J-th addendum on the right of Formulas (4,4.1); the convolution is by   28.3  and making the inverse transform 8(4) or 8.1(1), or using the integral kernel  as in § 28.5, one gets a solution   (D1,D4,i-vii) so that coefficients of an operator A (see 28(2


an original on U or a generalized function with the support if f is an original or a generalized function on U .Choose operators j A with constant coefficients on in the class of originals or generalized functions on U the problem (see 28(1 condition in accordance with 28(5.1).With any boundary conditions in the class of originals or generalized functions on additional borders \ .When the right side of 28(6) is non-trivial, thenj f is non-trivial.If 1 j f  is calculated,then the boundary conditions on\ the operator A j and the boundary conditions 28(5.1) on the boundary \ with the boundary conditions 28(5.1) on .U  the multiparameter non-commutative transform Cases may be, when either the left hyperplane , By a support of a generalized function g is called the family, denoted by gives only one or two additives on the right side of 21(1) in accordance with 21(3).
4, 6-8, 11, 13, 14, 16, 17, Proposition 10 and Corollary 4.1 are satisfied as well taking specific originals f with supports in V .At first take domains W which are quadrants, that is canonical closed subsets affine diffeomorphic with =1 then the addendum with ,1 are also used, where  denotes a real variable along a unit external normal to the boundary k b  for each k .Another example is: 1  is the real va-riable, while R is the center of the Cayley-Dickson algebra All coefficients a  are either constant or differentiable of the same class on each Operators depending on a less set 1 , , 1 (6,8)Thus from Relations(6,8)and 2(1,2,2.1,4 and Euler's formula one deduces expressions for