On the Fourier Spectra of Distributions in Clifford Analysis

Chin. Ann. Math.27B(5), 2006, 495- -506Chinese Annals ofDOI: 10.1007/s11401-006-0053-3Mathematics, Series BC The Editorial Ofice of CAM andSpringer- Verlag Berlin Heidelberg 2006On the Fourier Spectra of Distributionsin Clifford AnalysisFred BRACKX* Bram De KNOCK* Hennie De SCHEPPER*Abstract In recent papers by Brackx, Delanghe and Sommen, some fundamental higherdimensional distributions have been reconsidered in the framework of Clifford analysis,eventually leading to the introduction of four broad classes of new distributions in Euclideanspace. In the current paper we continue the in-depth study of these distributions, morespecifically the study of their behaviour in frequency space, thus extending classical resultsof harmonic analysis.Keywords Fourier spectra, Distributions, Cliford analysis2000 MR Subject Classifcation 30G35, 46F101 IntroductionDuring the last fty years, Cliford analysis has gained interest as a comprehensive functiontheory offering a direct, elegant and powerful generalization to higher dimension of the theoryof holomorphic functions in the complex plane. In its most simple but still useful setting,fat m-dimensional Euclidean space, Cliford analysis is centered around so-called monogenicfunctions, i.e. null solutions of the Cliford-vector valued Dirac operatora=Ee38x,where (e1,... , em) forms an orthogonal basis for the quadratic space Rm underlying the con-struction of the Cliford algebra Ro,m. Monogenic functions have a special relationship withharmonic functions of several variables in that they are refining their properties. Note for in-stance that each harmonic function can be split into a so-called inner and an outer monogenicfunction, and that a real harmonic function is always the real part of a monogenic one, whichdoes not need to be the case for a harmonic function of several cqmplex variables. The reason isthat, as does the Cauchy -Riemann operator in the complex plane, the rotation- invariant Diracoperator factorizes the m-dimensional Laplace operator. It hence is not surprising that Cliffordanalysis often leads to refinements or generalizations of classical results from harmonic analysis.In [3] and [4] four broad families of distributions in Euclidean space T.,p，Ux,p, V.,p andWx,p, depending on parametersλ∈C andp∈No= {0,1, 2,..}, were introduced and studiedin the framework of Cliford analysis. These distributiqne ol1 errina from theoroady cassically中国煤化工known distributionT\= Fpr李TYHCNMHGManuscript received January 26, 2006.*Ciford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University,GalgIaan 2, 9000 Gent, Belgium. E mail: fb@cage.UGent.be bdk@cage.UGent.be hds@cage.UGent.be496F. Brackx, B. De Knock and H. De Schepperdepending on the complex parameter 入. Here Fp is the fundamental distribution “initeparts" on the real line, and spherical co-ordinates have been used to convert an originallym-dimensional distribution into one acting on the real line. More precisely,(Tx,中) = QmFp/*,r)+m-1[[h J_ . 中)(S(@)]) = (r+)-1+0)101),m-1where am is the area of the unit sphere sm- 1 in IRm and 20[中] denotes the so-called sphericalmean of the testing function φ, obtained through integration over the unit sphere. In these,spherical co- ordinates not only reflect the“spherical" philosophy of the approach, encompassingall dimensions at once as opposed to a cartesian or tensorial approach with products of one-dimensional phenomena, but also enable to carry out the explicit calculations in one dimensionafter which they are exported again to the original setting of Euclidean space.An analogous approach underlies the definition of the four families of distributions men-tioned above, which respectively involve the inner spherical monogenics Pp(四) (p∈No), ie,restrictions to the unit sphere of monogenic polynomials which are vector valued and homogeneous of degree p, and the related outer spherical monogenics Pp(@)业,业Pp(四) and业Pp(四)4.Here, the spherical philosophy requires the introduction of generalized spherical means, wherethe involved spherical monogenic (inner or outer) appears in the integrand over the unit sphere.However, in view of the Fourier transformations aimed at, the distributions TA,p and their nor-malized versions T*。are now reconsidered from a cartesian point of view in Section 3 of thepaper.Any of the distributions in these families may be considered as a kernel (K) for a convolu-tion operator (L): L[f]= K*f. As is well known, see [20], such an operator may be realizedin frequency space by a multiplication operator, its so- clldl Fourier symbo: F[f]] = aF[f],where a = F[K]. This underlines the importance of calculating the Fourier transforms of thedistributions under consideration. It is worth mentioning that, for specifc values of the para-meters λ∈C and p∈No, those distributions turn into known kernel functions in harmonic andClifford analysis: up to constants, U_ m,0 reduces to Pv荒，the higher dimensional analogue ofthe so- called "Principal Value" distribution on the real line which constitutes the convolutionkernel for the higher dimensional Hilbert transform (see [5- 8]); U_ m+1,0 reduces to the funda-mental solution of the Dirac operator 2, while T_ m+2,0 is nothing but the fundamental solutionof the Laplace operator; furthermore, for λ = -p the inner and outer spherical monogenicsPp(4), Pp(@)4 and wP(4) are recovered; etc. Moreover when λ = -m- P the four fami-lies provide examples of so calld principal value distributions (see [13, 20]), being tempereddistributions obtained by a limiting process:(K,φ) = lim) JRM\B(0,e)K(x)中(x)dV()，φ∈ S(R").In [13, 20] much attention is paid to the calculation of the Fourier transforms of the principalvalue distributions where the function K takes the formK(x)=NC，出中国煤化工YHCNMHGwith k∈L2(Sm-1) such thatJ_k()dS(四) = 0.On the Fourier Spectru of Distributions497Such kind of principal value distributions lead to convolution operators in the following way:(K *中)(w)= lim_K(u- 正)中(国)dV(国), φ∈ S(R")JRm\B(0,e)also known as singular integral operators (see [20, Theorem VI.3.1]). In this paper, we concen-trate on the families Txp, Ux,p and Va,p introduced in [3, 4], for which we aim at calculatingthe Fourier spectra, thus generalizing the results obtained in [13, 20]. This is the subject ofSections 4 and 5.In order to make the paper self-contained we recall in Section 2 some basic notions andresults of Cifford analysis.2 Clifford AnalysisCliford analysis offers a function theory which is a higher dimensional analogue of the theoryof holomorphic functions of one complex variable. For more details concerning this functiontheory and its applications (for instance to harmonic analysis) we refer the reader to [2, 9, 11,12, 15 18].Let, for m≥2, RO,m be the real vector space Rm, endowed with a non-degenerate quadraticform of signature (0, m), let (e1,..，em) be an orthonormal basis for RO,m, and let Ro,m be theuniversal Cifford algebra constructed over RO,m. The non-commutative multiplication in Ro,mis then governed by the rules .e?=-1， i=1,2,.,m and eiej+eje;=0， i≠j.Foraset A={i,.. ,in}C {1,... ,m} with1≤i1 aaea， aA∈RAor still asa= 2[a]k where [a]k=. E aaeA is a so-called k-vector (k= 0,1,..,m). If wedenote the space of k-vectors by R谷, m, then Ro,m由R, leading to the identification of Rand RO,m with respectively Ro,m and Ro.m. We will also identify an elementr= (x1,... ,xm)∈R" with the one -vector (or vector for short)玉= E xj ej. The multiplication of any two vectors工andyisgivenbyy=-正,y +国^ywithm细业=2xBj=(工业+业),亚八y= 2 er(xs-xzy)=z(0,0，x<0,which is a regular distribution for Reμ> -1. In addition, one defines, for n∈N and μ∈Csuch that -n- 1 < Reμ < -n, the classical one -dimensional“inite part" distribution Fpx4by(P24,二/* ()- () -∞-..φn-)(0)2x"-1)dx1!(n-1)!e4+1φn-1)(0) eu+n \= lin(/, *咖2)l+()μ+ 1(n-1)! μ+n)As a function of pu, x哗is holomorphic in the half-plane Reμ > -1, and by analytic continuationFpx4 is holomorphic in C\{-1,-2, -3,-.}, the singular pointsμ= -n (n∈N) being simple(-1)-1-) . This finite part distribution shows the following properties:poles with residue i-D)T 8Yd二Fp叫=μFpx叫，μ≠0,-1,-2,-3,，.， xFpx件 = Fpx+，μ≠-1,-2,-3,...Remark 3.1 By a slight change in the above expression for Fpx件a definition may begiven for negative entire exponents as well, through the so-called monomial pseudofunctionsFpx7", n∈N (see e.g. [10, 19):ε-n+1φ(n-2)(0)e1、φ(n-1)(0)(Ppx功”, d(2》= im([.^x~"φ(x) dx+中(0)--n+ 1(n-2)! (-1)+ (n-1)!with properties中国煤化工n+1-FpxT"=(-n)Fpx7n-1+(-1)"= 58n!~YHCNMHGHowever, in what fllows, we have chosen to deal with the singularities of Fp x4 in anotherway.500F. Brackx, B. De Knock and H. De SchepperNext we define the generalized spherical mean 2p[φ] (see also [21]), for a scalar valuedtesting function 中(四) in Rm and a vector valued, monogenic, homogeneous polynomial Pp(x)of degree p≠0, as(i) 2%[中] = s(0{P2x(凹)中()]=;P2r(凹)中(工) dS(心),(i) 2(h>+|[@] = z(0)[r P2k+1()中([)] =。Pa+(西)中(国) dS(W).Am Jsm-1Finally we define the distributions Txp where λ∈C and p∈No, as follows. Let φ be ascalar valued testing function, let μ=λ+m-1 and put Pe=pifpis even, andpe=p-1 ifp is odd; then(Tx,p,$> = am(Fpr4+Pe , 2)[p).(3.1)Let, for a moment, λ≠-m-n and λ≠-m-n-Pe, n= 0,1,2,... Then the connectionbetween Tx, and T = T,0 is obtained in a natural way from(TPp(2),中(工)) = =- m-p-n+(Txp,) =am((n- 1)!(3.3)In this subsection we will examine these singularities more closely, revealing that in severalsubcases the residues turn out to be zero, on account of some specific properties of the gene-ralized spherical mean operator ip and of the polynomial Pp(x), respectively. Indeed, it hasbeen shown in [3] thatProposition 3.1 The spherical mean z[φ] is an even testing function on the real r- axcis.Its derivatives of odd order vanish at the origin r = 0, while for the derivatives of even orderwe have(21)!Pe +{8}'z[){}]}r=o =(Pe+ 20C(登+1)中国煤化工(3.4)with constantsYHCNMHG24!(mC()=2n(+1-1)...())，l∈No.On the Fourier Spectra of Distributions501In addition we may prove the following important results.Proposition 3.2 Let Pp() be a vector valued, monogenic, homogeneous polymomial ofdegreep and letr= |国. Then for eachl∈No,0，ifll. First, letl < p-暨. Then clearly, for eacha with |@| =p we havel< p- Sg: Invoking (3.10) we thus have that218...mr21=0，Va= (an,.. ,am), |&|=p,yielding Pp()r2l = 0. Next, takep-暨≤l< p. In this case, the arguments of Step 3 may berephrased quite literally, leading to an analogous result as in (3.11), however without the termforj= 0, since j will start fromp-l> 0. So, also here Pp(@)r2 = 0, implying that the secondpart of (3.5) holds.Step 5 Finally, expression (3.6) may be shown by conversion of (3.5) to frequency spaceand invoking properties (2.1) of the Fourier transform.The proof is completed.Returning to (3.3) for a more precise calculation of the residues, we will consider two distinctcases, according to the parity of n.Case A n=2l+2, l∈No.In this case we rewrite (3.3) asResλ=- -m-p-21-1(Tx,p,$) =(2l+1)!(2l+不the latter being zero on acount of Proposition 3.1. He-中国煤化ipoles whenevern= 2l + 2, or equivalently,入λ=-m-pe-2l-1,l∈Thus, the distributionsT-m -pe- -21 _1,p,l∈No canTYHc N M H Gmiting proces:(Tm-2-1,p,()=Qm lim_ (Ppr*,Q°)(p),On the Fourier Spectra of Distributions503where the limit at the right-hand side exactly yields the monomial pseudofunction Fpr+'2l-2(see Remark 3.1).Case B n=2l+ 1, l∈No.Substitution of these values of n in (3.3) yieldsRs--m-Po- (x,)= ((.5>29212)〉1= ampe + 201C(号+0(P(国)O譬+8(),中), .(3.12)the last step holding on account of (3.4). Since, according to (3.6), the expression at theright-hand side of (3.12) equals zero for p>暨+ l, we conclude that Tx,p also has no genuinesingularities in the case λ= -m-pe一2l forl= 0,1,2,..,p-登-1; for this finite set ofvalues, the distribution can be defined similarly as above by a limiting process, now involvingthe monomial pseudofunction Fpr2- 1 .The results obtained are summarized in the following theorem.Theorem 3.1 Considered as a function of (\,p)∈C x No, the distribution T。showssimple poles atλ=-m- 2p- 21, l∈No, with residueRes=-m- 2)-2Txp = am72p+ 20Cp+nP(四)4P+6(国).Remark 3.2 The above considerations lead to the conclusion that multiplication of Twith Pp(工) in (3.2) causes the removal of its singularities λ= -m- 2l forl < p. Hence, theequality (3.2) may be holomorphically extended to all couples (入, p) which do not fulfll therelation λ + 2p= -m- 2l, l∈No. This means that, whenever Txp is wll-defined, we mayrewriteit as T\ Pp.3.3 The distributions T,pIn [7] the distributions T* are defined as normalizations of the distributions T. This is doneby removing the singularities of T through the well-known technique of division by a delibe-rately chosen T-function. Here we generalize this normalization procedure to all distributionsT\,p.Noting that the function r(2+m+2) shows exactly the same simple poles as T\,p, withresiduesRes>=-m- -2p-2T1λ+m+2p2we are lead to the following definition of the so-called normalized distributions Tp:Tx,p| Tip=πt+pT(^m +p)λ≠-m- 2p- 2l,T.-m-2=-2,p=(-1)I!π罗-lPp(2)O+68(), l∈No, .2p+2(p+)!F(受+p+1)where, at the singularities of T\,p, the normalized distribution T*。is defined, up to constants,as the quotient of the residues involved.According to the results of the previous subsection中国煤化工uld be inter-preted in terms of the monomial pseudofunction Fpr+1MHCNMHG.n+1,n∈N,but λ≠-m-2p-2l,l∈No. Moreover, one can verify that for p = 0 this definition is inaccordance with the definition ofT = Tr,o in [7].504F. Brackx, B. De Knock and H. De Schepper4 The Fourier Spectra of the Distributions T。For the calculation of the Fourier spectra of the distributions T。，we will start from theclassical result (see [20, Theorem IV.4.1]): for those couples (\,p) ∈Cx No for which Reλ isrestricted to the strip -m- p< Reλ < -p, the following formula holdsF[I\P()(y)=i-Pπ-學一λ-,r(部+p),T_λ- -m-2pPp(y)r(-)or, following the results of the previous section,T(-会)lT-x-m-2p.p.(4.1)In [14, Lemma 2] it is shown that, by means of analytic continuation, the above formula alsoholds in the larger strip -m - 2p < Reλ < 0. However, both sides of (4.1) being meromorphicfunctions in the complex variable入, through analytic continuation that equality is valid in eachopen connected area containing the strip -m一p< Reλ< -p, and where the expression onboth sides exist. Singularities occur in (4.1) when λ= -m- 2p- 2l,l∈No, for the distributionat the left-hand side and when λ = 2l,l∈No, for the one at the right-hand side. Naturally, thesame singularities are also contained in the involved T-functions. Consequently, (4.1) is seen tohold for λ belonging to the set h, which is defined as8=C\({-m- 2p- 2l:l∈No}U{2l:l∈No}).This smoothens the path for the following fundamental result.Theorem 4.1 The Fourier transform of the distributions Ti.p is given byFT,J=i-PT*x-m-2p,，V(A,p)∈Cx No.Proof Three cases have to be distinguished.(i) λ∈∩On account of (4.1) we indeed haveπ去严切π一令FT,p]=;r(t +)FI,p]=i~P;r(一劲)T-λ-m- 2p,p=i-PT-x-m-2p,.(i) λ=-m-2p-2l,l∈NoExploiting the definition ofT*x- m -2p,p and the properties of the Fourier transform we arriveat(-1)4!π罗-lF[P(2)OP+'8(x)]Tm-2-2,21 = 24 +DT(受+p+0(-1)42π号-___()2(2)+pP()p0+2t.22+2(0 + D)r(弯+p+中国煤化工Asp≤p+ l, Proposition 3.2 leads to the desired resulYHCNMHGπ专+p+lF[T-m-2p- a,p)=i~p=F(晋+p+1)'p)"Pp(@) =iTInm.,(4.2)On the Fourier Spectra of DistributionsIn the above, we have used the notation ρ= |y| .(ii) λ=2l,l∈NoThis case directly follows by the action of the Fourier operator on (4.2):FIT,)(@)=P T-m-2p- a,(1)=-7T7m-2-2z.p(4).The proof is completed.5 The Fourier Spectra of the Distributions U,p and VpAlong with the family of distributions Txp also two other families Ux,p and Vaxp have beendefined, in which the higher dimensional“signum distribution”丝plays an important role (seee.g. [3, 4]). While recalling their respective definitions we directly introduce the correspondingnormalizations following the procedure used for the Tx,p in Section 3. To conclude the paper,the Fourier spectra of those normalizations U*,。and Vi, are calculated.For a scalar valued testing function中(工) in R", and a vector valued, monogenic, homoge-neous polynomial Pp(血) of degree p≠0, the generalized spherical means zj[中and zβ[φ]are defined as follows (see also [21]):(i)唱网= 2(0)uP2x(四)中()] =4P2x(4)()dS(@),(i) 2+[] = 2(0IruPxk+(@)中(@)] =;Jgm-14P2k+1(四)中()dS(), .i)骤刚= 2(0[Pz(4)业中(国) =P2r(1)中(x)dS(x),(iv)以[|] = s()[rP2k+()中()] =二P2k+1(四)四邮(])dS(2). .Note that for p= 0 and P(x) = 1 we have that z(3)[4] = s[4.The definition of the distributions Ux,p and Va,p then is similar to the one of the distributionsTx,p introduced above, however involving the newly introduced spherical means: .(i) (Uxp,p) = am(Fpr#+Pe , 29"[]),(i) (Vx,p,中) = am(Fpr4+Po, 2qp)[$)>. .Clearly, also these distributions show an infinite number of singularities, in view of which we willintroduce their normalizations at once. The modus operandi from Subsection 3.3 is adopted,leading to the following definitions, withl∈No:U\,p(Uxp =n"Htp;r(A+n1+p)λ≠-m-2p-2l-1,(-m-2-21- = 2+2+)1(号+p+1+1)Vx,p(Vp=πt++p=r(2+m+1+ p)'(-1)p+1!π号-l中国煤化工Vi-2p-2-1.0 = 2+2++()1()号+p-MYHCNMHGIn order to calculate the Fourier spectra of the normalized distributions U.p and Vtp,note that they are interrelated with the“mother" family T,p by the multiplication with the506F. Brackx, B. De Knock and H. De SchepperCoo-function国as well as by the action of the Dirac operator A (see [1, Propositions 5.2 and5.3]):λ+m + 2pλ+m+2pxTxp=2πUx+1,p, 2Tx,= XUX-1.p; T.,pg="Vi+1.p， T,p2=入V-1.p.Hence it sufices to combine Theorem 4.1 with the properties (2.1) of the Fourier transform, toarrive at the following elegant results:Proposition 5.1 The Fourier transform of the distributions Uxp, respectively Vip, is givenFIUpI=i-D-1U_x-m- 2.,，F[V;p]=i-D-lV*x-m-2,p”References[1] Brackx, F., De Knock, B., De Schepper, H. and Sommen, F., Distributions in Ciford Analysis: an Overview,Cliford Analysis and Applications, S.-L. Eriksson (ed.) (Proceedings of the Summer School, Tampere,August 2004), Tampere University of Technology, Institute of Mathematics, Research Report, No. 82, 2006,59-73.[2] Brackx, F, Delanghe, R. and Sommen, F, Clifford Analysis, Pitman Publishers, Boston-London-Melbourne,1982.[3] Brackx, F, Delanghe, R. and Sommen, F., Spherical means and distributions in Clifford analysis, Ad-vances in Analysis and Geormetry: New Developments Using Ciford Algebra, T. Qian, Th. HempAing, A.McIntosch and F. Sommen (eds.), Trends in Mathematics, Birkhauser, Basel, 2004, 65- -96.[4] Brackx, F, Delanghe, R. and Sommen, F, Spherical Means, Distributions and Convolution Operators inClifford Analysis, Chin. Ann. Math, 24B(2), 2003, 133-146.[5] Delanghe, R, Sorme remarks on the principal value kernel, Complex Var. Theory Appl, 47, 2002, 653- 662.[6] Brackx, F. and De Schepper, H, On the Fourier transform of distributions and diferential operators inCliford analysis, Complex Var. Theory Appl, 49(15), 2004, 1079-1091.[7] Brackx, F. and De Schepper, H, Hilbert-Dirac operators in Cliford analysis, Chin. Ann. Math, 26B(1),2005, 1-14. .[8] Brackx, F. and De Schepper, H, Convolution kernels in Clifford analysis: old and new, Math. MethodsAppl. Sci, 28(18), 2005, 2173 -2182.[9] Delanghe, R, Sommen, F. and Soueek, V., Clifford Algebra and Spinor-Valued Functions, Kluwer AcademicPubl, Dordrecht, 1992.[10] Gelfand, I. M. and Shilov, G. E, Generalized Functions, Vol. 1, Academic Press, New York, 1964.[11] Gilbert, J. and Murray, M, Clifford Algebra and Dirac Operators in Harmonic Analysis, Cambridge Univ.Press, Cambridge, 1991.[12] Girlebeck, K. and SproBig, W, Quaternionic and Ciford calculus for physicists and engineers, Mathemat-ical Methods in Practice, Wiley, Chichester, 1997.[13] Horvath, J,, Singular Integral Operators and Spherical Harmonices, Trans. Amer. Math. Soc, 82, 1950,52- 63.[14] Kou, K. 1I, Qian, T. and Sommen, F., Generalizations of Fueter's theorem, Methods and Appl. Anal, 9(2),2002, 273- -290.[15] Mitrea, M., Ciford Wavelets, Singular Integrals and Hardy Spaces, Lecture Notes in Mathematics, 1575,Springer Verlag, Berlin, 1994.16] Qian, T., Hempfing, Th., McIntosh, A. and Sommen, F, Advances in Analysis and Geometry: NewDevelopments Using Cifford Algebras, Birkhauser Verlag, Basel Boston Berlin, 2004.[17] Ryan, J. Basic Cliford analysis, Cubo Math. Educ, 2, 2000, 226 -256.[18] Ryan, J. and Struppa, D., Dirac Operators in Analysis, Addison Wesley Longman Ltd., Harlow, 1998.[19] Schwartz, L, Th6orie des Distributions, Hermann, Paris,中国煤化工[20] Stein, E. and WeiB, G., Introduction to Fourier Analysi:片C N M H Gton Univ. Pres,Princeton, 1971.[21] Sommen, F, Spin groups and spherical means, Clifford Algebras and Their Applications in MathematicalPhysics, JSR Chisholm and AK Common (eds.), Nato ASI Series, D. Reidel Publ. Co., Dordrecht, 1985.