On the adial deivative of the delta distibution Fed Backx, Fank Sommen & Jasson Vindas Depatment of Mathematical Analysis, Faculty of Engineeing and Achitectue, Ghent Univesity Depatment of Mathematics, Faculty of Sciences, Ghent Univesity Dedicated to ou co-autho Fank on the occasion of his 60th bithday. Abstact Possibilities fo defining the adial deivative of the delta distibution δ(x) in the setting of spheical coodinates ae exploed. This leads to the intoduction of a new class of continuous linea functionals simila to but diffeent fom the standad distibutions. The adial deivative of δ(x) then belongs to that new class of so-called signumdistibutions. It is shown that these signumdistibutions obey easy-to-handle calculus ules which ae in accodance with those fo the standad distibutions in R m. MSC: 46F05, 46F0, 5A66, 30G35 Intoduction Given a distibution T in R m expessed in spheical coodinates x = ω, = x, ω S m, S m being the unit sphee in R m, one may wonde if a meaning could be given to the actions T, T, ω T and the like, which, in pinciple, ae fobidden actions and thus a pioi not defined in the standad setting. Indeed, diffeentiation of distibutions is well defined with espect to the standad catesian coodinates x, x,..., x m and multiplication of a distibution is only allowed by a smooth function, a condition which, clealy, is not satisfied by the functions and ω. We will tackle this poblem fo one specific distibution: the delta distibution δ(x), and in the fist place concentate on a possible definition of its adial deivative δ(x). The delta distibution is pointly suppoted at the oigin, it is otation invaiant: δ(a x) = δ(x), A SO(m) it is even: δ( x) = δ(x) and it is homogeneous of ode ( m): δ(ax) = a m δ(x) So in a fist, naive, appoach, one could think of δ(x) as a distibution which emains pointly suppoted at the oigin, otation invaiant, even and homogeneous of degee ( m ). Tempoaily leaving aside the even chaacte, on the basis of the othe cited chaacteistics the distibution δ(x) should take the following fom: δ(x) = c 0 x δ(x) + + c m xm δ(x)
and it becomes immediately clea that this appoach is impossible since all distibutions appeaing in this decomposition ae odd and not otation invaiant, wheeas δ(x) is assumed to be even and otation invaiant. It could be that δ(x) is eithe the zeo distibution o is no longe pointly suppoted at the oigin, both possibilities being quite unacceptable. But anothe idea is that δ(x) is not a usual distibution anymoe, and the same fo δ(x) and ω δ(x). In Section 7 we will intoduce a new class of bounded linea functionals on an appopiate space of test functions, vey simila to but diffeent fom the standad distibutions, and we will show that the above cited thee distibutions belong to that new class. The delta distibution in catesian coodinates The delta distibution δ(x) D (R m ) is, quite natually, vey well known and fequently used in physics to model point souces in vaious field theoies. Let us summaize its popeties. It is a scala distibution defined by δ(x), ϕ(x) = ϕ(0), ϕ D(R m ) which is of finite ode zeo, with catesian deivatives given by s x j δ(x), ϕ(x) = ( ) s δ(x), s x j ϕ(x) = ( ) s { s x j ϕ(x)} x=0, j =,..., m In paticula the action of the Diac opeato = m j= e j xj esults into the vecto valued ditibution given by δ(x), ϕ(x) = δ(x), ϕ(x) = { ϕ(x)} x=0 Note that we ae using hee the basis vectos (e j, j =,..., m) of R m as Cliffod vectos, geneating the Cliffod algeba R 0,m, fo which e j =, e i e j = e i e j = e j e i = e j e i, e i e j = 0, i j =,..., m. Fo moe on Cliffod algebas we efe to e.g. [5]. In this way the Diac opeato, which may be seen as a Stein Weiss pojection of the gadient opeato (see e.g. [6]) and undelies the highe dimensional theoy of monogenic functions (see e.g. [3]), lineaizes the Laplace opeato: =, which is the Fische o Fouie dual to x = x, x being the Cliffod vecto vaiable x = m j= e j x j. The action of the Eule opeato E = x = m x j xj on the delta distibution eveals the latte s homogeneous chaacte: E δ(x) = ( m) δ(x), while the action of the (bivecto valued) angula momentum opeato Γ = x = j<k j= e j e k (x j xk x k xj ) leads to Γ δ(x) = 0. 3 The delta distibution in spheical coodinates Intoducing spheical coodinates: x = ω, = x, ω = m j= e jω j S m, the Diac opeato takes the fom = ω + ω
To give an idea how the angula diffeential opeato ω = m j= e j ωj looks like, we mention hee its explicit fom in dimension m = : ω = e θ θ and in dimension m = 3 : ω = e θ θ + e ϕ sin θ ϕ, whee the meaning of the angula coodinates θ and ϕ is staightfowad. The Eule opeato in spheical coodinates eads: E =, while the angula momentum opeato Γ takes the fom Γ = ω ω = ω ω. In Cliffod analysis (see e.g. [3]) this opeato Γ is mostly called the spheical Diac opeato and the fact that Γ δ(x) = 0 confims that the delta distibution δ(x), when expessed in spheical coodinates, only depends on the adial distance, in othe wods the delta distibution is spheically symmetic. Finally the Laplace opeato is witten in spheical coodinates as = + (m ) + whee = ω ω ω ω = ω ω is the Laplace Beltami opeato, containing only angula deivatives, which implies that δ(x) = 0 and so δ(x) = δ(x) + (m ) δ(x) Using the basic fomulae E δ(x) = ( ) δ(x) = ( m) δ(x) and ω δ(x) = 0 we will now establish the fomulae concening the legal actions on the delta distibution of two specific diffeential opeatos containing the adial deivative, viz. ( ) and ( ). Note that the ( latte opeato is, up to a constant facto, nothing else but the deivative with espect to : ) = Poposition. The actions of the opeatos ( ) and ( ) on the delta distibution ae well defined and it holds that ( ) δ(x) = ( ) (m + ) δ(x) and δ(x) = δ(x) Poof Applying twice the Eule opeato we obtain consecutively ( ) δ = m δ ( ) δ + ( ) δ = m δ ( ) δ = m(m + ) δ ( ) δ = m(m + ) δ ( ) δ = (m + ) δ + S with S = c 0 δ + m j= c j xj δ, since δ = (x + + x m)( x + + x m ) δ = m δ, δ = 0 and xj δ = 0. 3
On the othe hand, we have ( ) ( ) ( ) δ = ( m) δ ( ) δ + ( ) δ = ( m) ( ) δ ( ) δ = m + ( )δ ( ) δ = δ m + S Now invoking the Laplace opeato, we obtain ( ) δ = ( ) δ + (m ) δ ( ) = (m + ) δ + S + (m ) = δ + m + S fom which it follows that S = 0. ( δ ) m + S Remak. Note that the esults of Poposition ae consistent with the expession of the Laplace opeato in spheical coodinates since ( ) ( ) δ(x) + (m ) δ(x) = (m + ) δ(x) + (m )( ) δ(x) = δ(x) Poposition. The action of the opeato (ω ) on the delta distibution is well defined and it holds that m (ω ) δ(x) = δ(x) = e j xj δ(x) Poof The esult easily follows fom the spheical fom of the Diac opeato, taking into account that ω δ(x) = 0. Remak. The esult of Poposition, which may also been witten componentwise as j= (ω j ) δ(x) = xj δ(x), j =,..., m is consistent with the expession of the Laplace opeato in spheical coodinates since δ(x) = δ(x) = (ω + ω)(ω ) δ(x) = δ(x) ω (ω) δ(x) = δ(x) + (m ) δ(x) whee the well known esult: ω (ω) = m was taken into account. 4
By iteated application of the fomulae established in the Popositions and, the following esults ae obtained. Coollay. Fo all k N one has (ω ) k δ(x) = ( ) k k δ(x) = k k! (m + )(m + 3) (m + k ) k δ(x) (ω ) k+ δ(x) = ( ) k ω k+ δ(x) = k k! (m + )(m + 3) (m + k ) k+ δ(x) (ω ) k δ(x) = k+ δ(x) = k (ω ) δ(x) (ω ) k+ δ(x) = m + k + (k + ) k+ δ(x) = m + k + (k + ) k+ (ω ) δ(x) Coollay. Fo all k N one has ( ) k δ(x) = k k! k δ(x) = ( ) k k k! k δ(x) The fomulae obtained in Coollay may be genealized by consideing poducts of adial deivatives of the delta distibution and natual powes of the adial distance. The esults of the following Poposition 3 ae obtained by a staightfowad computation invoking the identities, with k l, (i) x l k δ(x) = (k)(k ) (k l + ) (ii) x l+ k δ(x) = (k)(k ) (k l) (iii) x l k+ δ(x) = (k)(k ) (k l + ) (iv) x l+ k+ δ(x) = (k)(k ) (k l + ) (m + k )(m + k 4) (m + k l) k l δ(x) (m + k )(m + k 4) (m + k l) k l δ(x) (m + k)(m + k ) (m + k l + ) k l+ δ(x) (m + k)(m + k ) (m + k l) k l δ(x) It tuns out that when the sum of the ode of the adial deivative and the powe of is even then the poduct is a well defined scala adial opeato, and when this sum is odd then the poduct is a well defined vecto opeato involving the function ω. Poposition 3. One has, with k l, (i) (ii) l k δ(x) = ( ) k+l k l (m + )(m + 3) (m + k ) (k l)! (m + k )(m + k 4) (m + k l) k l δ(x) ω l+ k δ(x) = ( ) k+l k l (m + )(m + 3) (m + k ) (k l )! (m + k )(m + k 4) (m + k l) k l δ(x) (iii) ω l k+ δ(x) = ( ) k+l k l (m + )(m + 3) (m + k ) (k l)! (m + k)(m + k ) (m + k l + ) k l+ δ(x) (iv) l+ k+ δ(x) = ( ) k+l+ k l (m + )(m + 3) (m + k ) (k l)! (m + k)(m + k ) (m + k l) k l δ(x) 5
4 Anothe attempt to define δ(x) Fist note that the fomula m (ω ) δ(x) = δ(x) = e j xj δ(x) j= obtained in Poposition cannot be used to define the adial deivative δ(x) of the delta distibution since multiplication of a distibution by ω is not allowed. But the fomula ( ) δ(x) = (m + ) ( ) δ(x) () obtained in Poposition, could offe a possibility to define δ(x). Indeed, by taking squae oots we obtain m + H δ(x) = ( ) δ(x) () Let us explain this esult in moe detail. At the ight hand side of () appeas the so called squae oot of the negative Laplace opeato ( ) which is the convolution opeato given, fo an appopiate function o distibution F (x), by ( ) [F (x)] = Fp F (x) a m+ m+ m+ π Γ( m+ the convolution kenel Fp being a Finite Pat distibution in R m, and a m+ m+ = being ) the aea of the unit sphee S m in R m+. Repeated action of the opeato ( ) esults into the well known esult ( ) ( ) ( ) [F ] = Fp [Fp F ] = ( )[F ] a m+ m+ m+ ecoveing in this way fo F (x) = δ(x), up to the constant (m + ), the ight hand side of (). Note that the ight hand side of () educes to m + ( ) δ(x) = a m+ m + Fp m + m+ δ = Γ( m+ ) Fp π m+ m+ At the left hand side of () appeas the one dimensional Hilbet tansfom H in the eal vaiable, given, fo an appopiate function o distibution f(), by H [f()] = π Pv f the Hilbet kenel Pv being a Pincipal Value distibution in R. As is well known the one dimensional Hilbet tansfom is a linea endomophism both of the space D Lp of test functions given by D Lp = {φ(t) C (R) : φ (k) (t) L p (R), k N}, < p < + and of its dual D L p, with invese H = H, so that H =. Moeove the Hilbet tansfom is commuting with deivation so that epeated action of H leads to (H )(H ) δ(x) = H δ(x) = ( ) δ(x) 6
in this way ecoveing the left hand side of (). Now ewiting () as m + (H ) δ(x) = ( ) m + δ(x) = Γ( m+ ) Fp π m+ m+ (3) it becomes clea that the opeato (H ) acting on the delta distibution flattens out the point suppot of δ(x) to the whole of R m, and the esult ( ) δ(x) is the only distibution, up to a constant, which is otation invaiant and homogeneous of degee ( m ). Note by the way that the same phenomenon concening the suppot of the delta distibution occus unde the action of the so called Hilbet Diac opeato (H ) (see e.g. []): (H )[δ(x)] = ( ) δ(x) = Γ( m+ ) H[f] = H f = π m+ Fp m+ whee H stands fo the Hilbet tansfom in R m given, fo an appopiate function o distibution f, by m+ Γ( ) π m+ Fp ω m f In ode to etieve fom (3) an expession fo δ(x), we can act with the adial Hilbet tansfom H on both sides, leading to m + Γ( m+ δ(x) = ) [ H π m+ Fp ] m+ (4) which equies an appopiate definition of the adial Hilbet tansfom of a distibution and of a otation invaiant distibution in paticula, a question which is of the same natue as the quest fo an acceptable definition of the adial deivative δ(x). Anyway, as we will show that δ(x) belongs to a new class of so called signumdistibutions, by (4), the adial Hilbet tansfom of a distibution should belong to that class too. 5 A physics appoach to the delta distibution In physics texts one often encountes the following expession fo the delta distibution in spheical coodinates: δ(x) = δ() a m m (5) Appaently this can be explained in the following way. Wite the action of the delta distibution as an integal: ϕ(0) = δ(x), ϕ(x) = δ(x) ϕ(x) dv (x) R m = m δ(x) d ϕ( ω) ds ω 0 S m = a m m δ(x) Σ 0 [ϕ]() d using the so called spheical mean of the test function ϕ given by Σ 0 [ϕ]() = ϕ( ω) ds ω a m 7 0 S m
whee a m = π m Γ( m ) is the aea of the unit sphee Sm in R m. As it is easily seen that Σ 0 [ϕ](0) = ϕ(0) it follows that a m m δ(x) Σ 0 [ϕ]() d = δ(), Σ 0 [ϕ]() = 0 which explains (5). Howeve we pefe to intepet this expession as 0 δ() Σ 0 [ϕ]() d ϕ(0) = δ(x), ϕ(x) = δ(), Σ 0 [ϕ]() = Σ 0 [ϕ](0) (6) which can be genealized to highe even ode Diac deivatives of the delta distibution: { l ϕ(x)} x=0 = l δ(x), ϕ(x) = ( ) l C(l) l δ(), Σ 0 [ϕ]() = ( ) l C(l) { l Σ 0 [ϕ]()} =0 with C(l) = l l! ( m ) ( m ) (l)! + l = l l! Γ( m + l) (l)! Γ( m ), l = 0,,,... Note that the spheical mean Σ 0 [ϕ]() is an even function of, whose odd ode deivatives vanish at the oigin: l+ δ(), Σ 0 [ϕ]() = { l+ Σ 0 [ϕ]()} =0 = 0 Fo expessing the highe odd ode Diac deivatives of the delta distibution in a simila way we have to invoke the so called spheical mean of the second kind Σ [ϕ], which was intoduced in []: Σ [ϕ]() = ω ϕ( ω) ds ω a m S m This spheical mean Σ [ϕ]() is a vecto valued odd function of, whose even ode deivatives vanish at the oigin: l δ(), Σ [ϕ]() = { l Σ [ϕ]()} =0 = 0 It holds that o l+ δ(x), ϕ(x) = ( ) l C(l + ) l+ δ(), Σ [ϕ]() (7) { l+ ϕ(x)} x=0 = ( ) l C(l + ) { l+ Σ [ϕ]()} =0 In the physics language these esults would then be witten as l δ(x) = ( ) l C(l) δ (l) () a m m l+ δ(x) = ( ) l C(l + ) δ (l+) () a m m ω which, by means of the esults of Coollay, lead to the expessions l δ(x) = ω l+ δ(x) = δ (l) () (m)(m + ) (m + l ) (l)! a m m δ (l+) () (m)(m + ) (m + l) (l + )! a m m ω Again δ(x) escapes fom this appoach. 8
6 Spheical epesentation of a distibution When expessing a test function ϕ(x) D(R m ) in spheical coodinates, one obtains a function ϕ(, ω) = ϕ(ω) D(R S m ), but it is clea that not all functions ϕ(, ω) D(R S m ) stem fom a test function in D(R m ). Howeve a one to one coespondence may be established between the usual space of test functions D(R m ) and a specific subspace of D(R S m ). Lemma. (see [4]) Thee is a one to one coespondence ϕ(x) ϕ(, ω) = ϕ(ω) between the spaces D(R m ) and V = {φ(, ω) D(R S m ) : φ is even, i.e. φ(, ω) = φ(, ω), and { n φ(, ω)} =0 is a homogeneous polynomial of degee n in (ω,..., ω m ), n N}. Clealy V is a closed (but not dense) subspace of D(R S m ) and even of D E (R S m ), whee the suffix E efes to the even chaacte of the test functions in that space, and V is endowed with the induced topology of D(R S m ). The one to one coespondence between the spaces of test functions D(R m ) and V tanslates into a one to one coespondence between the standad distibutions T D (R m ) and the bounded linea functionals in V ; this coespondence is given by T (x), ϕ(x) = T (, ω), ϕ(, ω) By Hahn Banach s theoem the bounded linea functional T (, ω) V may be extended to the distibution T(, ω) D (R S m ); such an extension is called a spheical epesentation of the distibution T (see e.g. [7]). As the subspace V is not dense in D(R S m ), the spheical epesentation of a distibution is not unique, but if T and T ae two diffeent spheical epesentations of the same distibution T, thei estictions to V coincide: T (, ω), ϕ(, ω) = T (, ω), ϕ(, ω) = T (, ω), ϕ(ω) = T (x), ϕ(x) Fo test functions in D(R S m ) the spheical vaiables and ω ae odinay vaiables, and thus smooth functions. It follows that fo distibutions in D (R S m ) multiplication by and by ω and diffeentiation with espect to and to ω ae standad well defined opeations, and so T(, ω), Ξ(, ω) = T(, ω), Ξ(, ω) fo all test functions Ξ(, ω) D(R S m ), and simila expessions fo ω T, T and ω T. Howeve if T and T ae two diffeent spheical epesentations of the same distibution T D (R m ), then, upon esticting to test functions ϕ(, ω) V, we ae stuck with T (, ω), ϕ(, ω) T (, ω), ϕ(, ω) since ϕ(, ω) is an odd function in the vaiables (, ω) and does no longe belong to V (and neithe do ω ϕ(, ω), ϕ(, ω) and ω ϕ(, ω)). The conclusion is that the concept of spheical epesentation of a distibution does not allow fo an unambiguous definition of the actions poposed, confiming ou statement that the solution of ou poblem lays outside the wold of taditional distibutions. At the same time it becomes clea why the actions of the opeatos, and ω on a standad distibution ae well-defined instead. Indeed, we have e.g. T(, ω), Ξ(, ω) = T(, ω), Ξ(, ω) whee now Ξ(, ω) belongs to D E (R S m ) which enables estiction to test functions in V in an unambiguous way. 9
7 An altenative class of distibutions As aleady emaked in the peceding section, ω is an odinay (vecto) vaiable in R S m, whence it makes sense to conside the following subspace of vecto valued test functions in R S m : W = ω V D O (R S m ; R m ) D(R S m ; R m ) whee now the suffix O efes to the odd chaacte of the test functions unde consideation, i.e. ψ(, ω) = ψ(, ω), ψ D O (R S m ; R m ). This space W is endowed with the induced topology of D(R S m ; R m ). By definition thee is a one to one coespondence between the spaces V and W. Fo each U(, ω) D (R S m ; R m ) we define Ũ(, ω) W by the estiction Ũ(, ω), ω ϕ(, ω) = U(, ω), ω ϕ(, ω), ω ϕ(, ω) W In R m we conside the space Ω(R m ) = {ω ϕ(x) : ϕ(x) D(R m )}. Clealy the functions in Ω(R m ) ae no longe diffeentiable in the whole of R m, since they ae not defined at the oigin due to the function ω = x x. By definition thee is a one to one coespondence between the spaces D(Rm ) and Ω(R m ). Fo each Ũ(, ω) W we define U(x) by U(x), ω ϕ(x) = Ũ(, ω), ω ϕ(, ω)), ω ϕ(x) Ω(Rm ) Clealy U(x) is a bounded linea functional on Ω(R m ), which we call a signumdistibution. Now stat with a standad distibution T (x) D (R m ) and let T(, ω) D (R S m ) be one of its spheical epesentations. Put S(, ω) = ω T(, ω) which in its tun leads to the signumdistibution S(x) Ω(R m ). Then we consecutively have S(x), ω ϕ(x) = S(, ω), ω ϕ(, ω) = ω T(, ω), ω ϕ(, ω) = T(, ω), ϕ(, ω) = T (x), ϕ(x) since ω =, and we call S(x) a signumdistibution associated to the distibution T (x) and denote it by ω T (x). It should be emphasized that fo a given distibution T (x) the associated signumdistibution ω T (x) is not uniquely defined but instead depends on the spheical epesentation of T (x) chosen; moeove ω T is a mee notation, not the poduct of T and ω. Example. A locally integable function f(x) L loc (R m ) gives ise to a egula distibution T f via T f, ϕ(x) = f(x) ϕ(x) dx, ϕ(x) D(R m ) R m As ω = it is clea that also f(x) ω belongs to L loc (R m ), thus geneating the egula distibution T fω via T fω, ϕ(x) = f(x) ω ϕ(x) dx, ϕ(x) D(R m ) R m Howeve the same integals ae defining the egula signumdistibutions U fω and U f by U fω, ω ϕ(x) = f(x) ϕ(x) dx, ω ϕ(x) Ω(R m ) R m 0
f(x) ω ϕ(x) dx, ω ϕ(x) Ω(R m ) and U f, ω ϕ(x) = R m A spheical epesentation of T f and T fω espectively is given by T f (, ω), Ξ(, ω) = 0 m d f(ω) Ξ(, ω) ds ω S m T fω (, ω), Ξ(, ω) = 0 m d f(ω) ω Ξ(, ω) ds ω S m since esticting to the space V leads to T f (, ω), ϕ(, ω) = 0 m d f(ω) ϕ(ω) ds ω = T f, ϕ(x) S m and T fω (, ω), ϕ(, ω) = 0 m d f(ω) ω ϕ(ω) ds ω = T fω, ϕ(x) S m These paticula spheical epesentations induce signumdistibutions associated to T f (x) and T fω (x), which we define to be ω T f (x) and ω T fω (x) espectively. It thus holds that ω T f (x), ω ϕ(x) = T f (x), ϕ(x) = f(x) ϕ(x) dx and ω T fω (x), ω ϕ(x) = T fω (x), ϕ(x) = Clealy ω T f = U fω and ω T fω = U f. R m R m f(x) ω ϕ(x) dx Example. Conside the distibution T (x) = δ(x). Ou aim is to define the signumdistibutions ω δ(x), δ(x) and δ(x). A spheical epesentation of the delta distibution is given by T(, ω), Ξ(, ω) = Σ 0 [Ξ(, ω)]} =0 Indeed, when esticting to the space V and taking into account popety (6) we obtain T(, ω), ϕ(, ω) = Σ 0 [ϕ( ω)]} =0 = δ(x), ϕ(x) This paticula spheical epesentation of T (x) induces a signumdistibution associated to δ(x), which we define to be ω δ(x). It thus holds that ω δ(x), ω ϕ(x) = δ(x), ϕ(x) (8) Now conside the distibution T (x) = (ω ) δ(x) = δ(x) (see Poposition ). epesentation of this distibution T (x) is given by A spheical T (, ω), Ξ(, ω) = m { Σ [Ξ(, ω)]} =0 Indeed, when esticting to the space V and taking into account popety (7) we obtain T (, ω), ϕ(, ω) = m { Σ [ϕ( ω)]} =0 = δ(x), ϕ(x)
This paticula spheical epesentation of T (x) induces a signumdistibution associated to T (x) = (ω ) δ(x), which we define to be δ(x). It thus holds that δ(x), ω ϕ(x) = (ω ) δ(x), ϕ(x) (9) Finally as x δ(x) = 0, we define the signumdistibution δ(x) to be the zeo signumdistibution. Example 3. We conside the distibution δ(x) = (ω ) δ(x). As x δ(x) = m δ(x), we fist define, on the basis of a simila easoning as in the pevious example, the signumdistibution δ(x) by δ(x), ω ϕ(x) = m δ(x), ϕ(x) In view of (8) it clealy holds that δ(x) = ( m) ω δ(x) o Moe geneally, by consideing the distibution we define the signumdistibution k+ δ(x) by (ω ) δ(x) = ( m) ω δ(x) (0) x k+ δ(x) = (m + k) k δ(x) k+ δ(x), ω ϕ(x) = (m + k) k δ(x), ϕ(x) o, invoking the fomulae obtained in Coollay, (ω k+ ) δ(x), ω ϕ(x) = (m + k) k δ(x), ϕ(x) Now we define the signumdistibution ω δ(x) = ω (ω ) δ(x) to be given by In view of (9) it clealy holds that ω δ(x), ω ϕ(x) = δ(x), ϕ(x) ω (ω ) δ(x) = δ(x) Moe geneally we define the signum distibution ω (ω k+ ) δ(x) by ω (ω k+ ) δ(x), ω ϕ(x) = (ω k+ ) δ(x), ϕ(x) Finally we define the signumdistibution (ω k+ ) δ(x) by (ω k+ ) δ(x), ω ϕ(x) = k+ δ(x), ϕ(x) Example 4. We conside the distibution δ(x) = δ(x). As x δ(x) = δ(x), we fist define the signumdistibution (x) by δ(x), ω ϕ(x) = δ(x), ϕ(x) Clealy δ(x) = δ(x), fom which it also follows, by means of the esults in Poposition, that ( ) δ(x) = δ(x) and that ( ) δ(x) = (m + ) δ(x) ()
Moe geneally, based upon the fomula x k δ(x) = (k) k δ(x), we define the signumdistibution k δ(x) by k δ(x), ω ϕ(x) = (k) k δ(x), ϕ(x) o, again invoking the fomulae obtained in Coollay, k δ(x), ω ϕ(x) = (m + k ) (ω k ) δ(x), ϕ(x) Now we define the signumdistibution ω k δ(x) by o ω k δ(x), ω ϕ(x) = k δ(x), ϕ(x) ω k δ(x), ω ϕ(x) = k δ(x), ϕ(x) Finally we define the signumdistibution k δ(x) by which tuns into o still k δ(x), ω ϕ(x) = (ω ) k δ(x), ϕ(x) k δ(x), ω ϕ(x) = (ω ) k δ(x), ϕ(x) k+ δ(x), ω ϕ(x) = (ω k+ ) δ(x), ϕ(x) Remak 3. Fo k = 0 we obtain in paticula that δ(x) = 0. In fact this poduct is defined within the famewok of standad distibutions since the delta distibution is of finite ode zeo and the function is continuous in R m. Example 5. Division of a standad distibution by a smooth function being allowed, we have x δ(x) = m δ(x) + S 0 whee S 0 stands fo δ(x) c 0 with c 0 an abitay constant vecto, since x δ(x) = m δ(x) and x S 0 = 0. Making use of this fomula we define the signumdistibution δ(x) by In view of and δ(x), ω ϕ(x) = x δ(x), ϕ(x) = m δ(x) S 0, ϕ(x) δ(x), ω ϕ(x) = δ(x), ϕ(x) ω S 0, ω ϕ(x) = S 0, ϕ(x) we obtain the following elation involving signumdistibutions: δ(x) = m δ(x) + ω δ(x) c 0 But as we expect the signumdistibutions δ(x) and δ(x) to be SO(m) invaiant, the abitay vecto constant c 0 should be zeo and we end up with δ(x) = m δ(x) 3
Moe geneally we put k+ δ(x), ω ϕ(x) = ( )k+ δ(x), ϕ(x) xk+ = ( ) k+ k k!(m + k)(m + k ) (m) k+ δ(x) + S k, ϕ(x) with S k an abitay vecto linea combination of deivatives of the delta distibution up to ode k, which eventually leads to (m )! δ(x) = k+ (m + k)! k+ δ(x) Also, based upon the following fomulae fo division in catesian coodinates: x k+ δ(x) = x k δ(x) = k + k+ δ(x) + S 0 m + k k+ δ(x) + S 0 and in a simila way as above, it is shown that and (ω k+ )δ(x) = m + k + (ω k+ )δ(x) + ω δ(x)c 0 k δ(x) = m + k k+ δ(x) 8 Some calculus Thee is a fundamental sequence of deivatives of the delta distibution, which ae altenatively scala and vecto valued, geneated by the action of the opeato (ω ): δ (ω ) δ δ... (ω k ) δ k δ (ω k+ ) δ... and fo each of the distibutions in this sequence we have, in the examples of the foegoing section, defined, though the action of ω, a specific associated signumdistibution, yielding in this way a paallel sequence of signumdistibutions: ω δ δ (ω ) δ... k δ (ω k ) δ k+ δ... Let us ecall these definitions. The initial definition is the following. Definition. ω k δ(x), ω ϕ(x) = k δ(x), ϕ(x) () ω (ω k δ(x)), ω ϕ(x) = (ω k ) δ(x), ϕ(x) (3) Wheeupon we intoduce the signumdistibution k δ(x) by Definition. k δ(x) = ω (ω k δ(x)) 4
such that definition (3) may be ephased as k δ(x), ω ϕ(x) = (ω k ) δ(x), ϕ(x) (4) Thee ae two othe actions on each of the distibutions of the fist sequence yielding a signumdistibution of the second sequence, viz. the actions by and by. Indeed, by an appopiate combination of the above definitions, we obtain the following calculus ules. Popety. One has (ω k+ ) δ(x) = (m + k) ω k δ(x) k δ(x) = (m + k ) k δ(x) (ω k+ ) δ(x) = ω k+ δ(x) k δ(x) = k+ δ(x) One may wonde if thee ae actions tansfoming the signumdistibutions fom the second sequence back into distibutions fom the fist sequence and the answe is positive. Indeed, the same actions apply on the signumdistibutions fom the second sequence. The basic action is again though the opeato ω, which yields the following definitions. Definition 3. ω (ω k ) δ(x), ϕ(x) = ω k δ(x), ω ϕ(x) (5) ω ( k δ(x)), ϕ(x) = k δ(x), ω ϕ(x) (6) Compaing the definitions (5) and () it is clea that the distibution ω (ω k else but the distibution k ) δ(x) is nothing δ(x), while compaing definitions (6) and (4) shows that the dis- ) δ(x). tibution ω ( k δ(x)) is indeed the distibution (ω k Fo the actions of the opeatos and on the signumdistibutions, which ae defined in a simila way as the actions of and on distibutions, we obtain the following computation ules. Popety. One has k+ δ(x) = (m + k) k δ(x) (ω k ) δ(x) = (m + k ) (ω k ) δ(x) ( ) k δ(x) = k δ(x) ( ) ω k+ δ(x) = ω k+ δ(x) ( k+ ) δ(x) = k+ δ(x) (ω k ) δ(x) = (ω k+ ) δ(x) This leads to the following completely symmetic pictue... (ω k ) δ k δ (ω k+ ) δ... ω ω ω ω ω ω... k δ (ω k ) δ k+ δ... 5
Remak 4. When composing two opeatos out of, and ω, six opeatos oiginate:,, ω,, ω and ω, which ae taditional opeatos whose actions on distibutions ae well defined. This means that the consecutive action by any two of the opeatos, and ω should lead to a known esult, which is a seious test fo all calculus ules established above. We now pove that this is indeed the case. (i) By the calculus ules we have and (ω k+ ( k δ) = (m + k)(ω k δ) = (m + k )( k On the othe hand, invoking the identities and the fomulae of Coollay we have and δ) = (m + k )(m + k)(ω k ) δ δ) = (m + k )(m + k ) k δ x k+ δ(x) = (m + k)(k) k δ(x) (7) x k δ(x) = (m + k )(k) k δ(x) (8) (ω k+ δ) = ( )k k (m + )(m + 3) (m + k ) x k+ δ k! = (m + k )(m + k)(ω k ) δ ( k δ) = ( )k k (m + )(m + 3) (m + k ) x k δ k! = (m + k )(m + k )(ω k ) δ (ii) The Eule opeato measues the degee of homogeneity and thus while the calculus ules lead to (iii) By the calculus ules we obtain (ω k+ δ) = (m + k + ) ω k+ δ ( k δ) = (m + k) k δ (ω k+ δ) = (ω k+ δ) = (m + k + ) ω k+ ( k δ) = ( k+ δ) = (m + k) k δ ω (ω k+ δ) = k+ δ = (m + k) k δ ω ( k δ) = (m + k ) ω k δ while invoking the identities (7) and (8) espectively leads to ω (ω k+ δ) = ( )k k (m + )(m + 3) (m + k ) x k+ δ k! = (m + k) k δ δ 6
and (iv) The calculus ules lead to ω ( k δ) = ( )k k (m + )(m + 3) (m + k ) x k δ k! = (m + k ) ω k δ (ω k+ δ) = ω k+3 δ ( k δ) = k+ δ On the othe hand we can make use of the identities to obtain (ω ) k δ(x) = k+ δ(x) (ω ) k+ δ(x) = m + k + k+ δ(x) (k + ) (ω k+ δ) = ( )k k (m + )(m + 3) (m + k ) ( ) m + k + k+3 δ k! (k + ) and = ( ) k+ k+ (k + )! (m + )(m + 3) (m + k )(m + k + ) k+3 δ = ω k+3 δ ( k δ) = ( )k k (m + )(m + 3) (m + k ) ( ) m + k + k+ δ k! (k + ) = ( ) k+ k+ (k + )! (m + )(m + 3) (m + k )(m + k + ) k+ δ = k+ δ (v) On the one hand we have by the calculus ules and on the othe ω (ω k+ δ) = ω (ω k+ δ) = k+ ω ( k δ) = ω ( k+ δ) = ω k+ δ ω (ω k+ δ) = ω ( ) k (ω ) k+ δ = ( ) k (ω ) k+ δ = k+ δ ω ( k δ) = ω ( ) k (ω ) k δ = ( ) k (ω ) k+ δ = ω k+ δ (vi) The action by ω = is tivial. 9 Conclusion In ou quest fo an unambiguous meaningful definition, in the famewok of spheical coodinates, of the adial deivative δ(x) of the delta distibution in R m, and of othe fobidden actions on the delta distibution such as δ(x) and ω δ(x), we wee faced with the impossibility to achieving this within the familia setting of the taditional distibutions. Instead we had to intoduce a new space of continuous linea functionals on a space of test functions showing a singulaity at the oigin, fo which we coined the tem signumdistibutions, beaing in mind that ω = x x may be δ 7
intepeted as the highe dimensional countepat to the signum function on the eal line. It tuns out that the actions by, ω and map a distibution to a signumdistibution and vice vesa, and a numbe of efficient calculus ules fo handling these tansitions fo the delta distibution wee established. As the composition of any two opeatos fom, ω and esults in a legal and well defined action on distibutions, and on the delta distibution in paticula, these calculus ules wee positively tested fo this phenomenon. Finally it should be mentioned that spaces of test functions showing a singulaity at the oigin also appea in the theoy of so called thick distibutions (see [8]). The possible elationships between the signumdistibutions and the thick distibutions ae subject of cuent eseach. 0 Acknowledgement The fist autho wants to thank Kevin Coulembie, Hendik De Bie, Hennie De Scheppe, and David Eelbode fo thei inteest in and thei valuable comments on the topic teated in this pape. Refeences [] F. Backx, R. Delanghe, F. Sommen, Spheical Means and Distibutions in Cliffod Analysis, Tends in Mathematics: Advances in Analysis and Geomety, Bikhäuse Velag (Basel, 004), 65 96. [] F. Backx, H. De Scheppe, Hilbet Diac Opeatos in Cliffod Analysis, Chin. Ann. Math. Se. B 6 () (005), 4. [3] R. Delanghe, F. Sommen, V. Souček, Cliffod Algeba and Spino Valued Functions: A Function Theoy fo the Diac Opeato, Kluwe Academic Publishes (Dodecht, 99). [4] S. Helgason, The Radon tansfom, Bikhäuse (Boston, MA, 999). [5] I. Poteous, Cliffod Algebas and the Classical goups, Cambidge Univesity Pess (Cambidge, 995). [6] E. M. Stein, G. Weiss, Genealization of the Cauchy-Riemann equations and epesentations of the otation goup, Ame. J. Math. 90 (968), 63 96. [7] D. Vučković, J. Vindas, Rotation invaiant ultadistibutions, in: Genealized Functions and Fouie Analyis, Opeato Theoy: Advances and Applications, Spinge (Basel, 07) (to appea). [8] Y. Yang, R. Estada, Distibutions in spaces with thick points. J. Math. Anal. Appl. 40 (03), 8 835. 8