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Contents

Editorial Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

R. Abreu-Blaya, J. Bory-Reyes and T. Moreno-GarcaTeodorescu Transform Decomposition of Multivector Fieldson Fractal Hypersurfaces

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1. Cliord algebras and multivectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2. Cliord analysis and harmonic multivector elds . . . . . . . . . . . . . . . . 42.3. Fractal dimensions and Whitney extension theorem . . . . . . . . . . . . . 5

3. Jump problem and monogenic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64. K-Multivectorial case. Dynkin problem and harmonic extension . . . . . . 95. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.1. The curve of B. Kats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2. The surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3. The function u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.4. Proof of properties a) e) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

F. Brackx, N. De Schepper and F. SommenMetric Dependent Cliord Analysis with Applications to Wavelet Analysis

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172. The metric dependent Cliord toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1. Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2. From Grassmann to Cliord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3. Embeddings of Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4. Fischer duality and Fischer decomposition . . . . . . . . . . . . . . . . . . . . . . 322.5. The Euler and angular Dirac operators . . . . . . . . . . . . . . . . . . . . . . . . . 362.6. Solid g-spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.7. The g-Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3. Metric invariant integration theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.1. The basic language of Cliord dierential forms . . . . . . . . . . . . . . . . 49

vi Contents

3.2. Orthogonal spherical monogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2.1. The Cauchy-Pompeiu formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2.2. Spherical monogenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4. The radial g-Cliord-Hermite polynomialsand associated CCWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1. The radial g-Cliord-Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . 594.2. The g-Cliord-Hermite wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3. The g-Cliord-Hermite Continuous Wavelet Transform . . . . . . . . . . 63References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

P. Dewilde and Sh. ChandrasekaranA Hierarchical Semi-Separable Moore-Penrose Equation Solver

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692. HSS representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734. HSS row absorption procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Complexity calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785. An HSS Moore-Penrose reduction method . . . . . . . . . . . . . . . . . . . . . . . . . . . 796. Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Methods from Multiscale Theory and Wavelets Applied to Nonlinear Dynamics

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872. Connection to signal processing and wavelets . . . . . . . . . . . . . . . . . . . . . . . . 883. Motivating examples, nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

MRAs in geometry and operator theory . . . . . . . . . . . . . . . . . . . . . . . . 923.1. Spectrum and geometry: wavelets, tight frames, and

Hilbert spaces on Julia sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.1.2. Wavelet lters in nonlinear models . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.2. Multiresolution analysis (MRA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.2.1. Pyramid algorithms and geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.3. Julia sets from complex dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.1. Spectral decomposition of covariant representations:

projective limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075. Remarks on other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

D.E. Dutkay and P.E.T. Jorgensen

Contents vii

K. GustafsonNoncommutative Trigonometry

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

2. The rst (active) period 19661972 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

3. The second (intermittent) period 19731993 . . . . . . . . . . . . . . . . . . . . . . . . . 131

4. The third (most active) period 19942005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5. Related work: Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6. Noncommutative trigonometry: Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.1. Extensions to matrix and operator algebras . . . . . . . . . . . . . . . . . . . . . 1436.2. Multiscale system theory, wavelets, iterative methods . . . . . . . . . . . 1466.3. Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

H. HeyerStationary Random Fields over Graphs and Related Structures

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

2. Second-order random elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1582.1. Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1582.2. Spatial random elds with orthogonal increments . . . . . . . . . . . . . . . 1592.3. The Karhunen representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

3. Stationarity of random elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633.1. Graphs, buildings and their associated polynomial structures . . . 1633.1.1. Distance-transitive graphs and Cartier polynomials . . . . . . . . . . . 1633.1.2. Triangle buildings and Cartwright polynomials . . . . . . . . . . . . . . . 1643.2. Stationary random elds over hypergroups . . . . . . . . . . . . . . . . . . . . . . 1653.3. Arnaud-Letac stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

M.W. Wong and H. ZhuMatrix Representations and Numerical Computations of Wavelet Multipliers

1. Wavelet multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

2. The Landau-Pollak-Slepian operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

3. Frames in Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4. Matrix representations of wavelet multipliers . . . . . . . . . . . . . . . . . . . . . . . . . 179

5. Numerical computations of wavelet multipliers . . . . . . . . . . . . . . . . . . . . . . . 180

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

viii Contents

J. Zhao and L. Peng

than 2-dimensional Euclidean Group with Dilations

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1832. Cliord algebra-valued admissible wavelet transform . . . . . . . . . . . . . . . . . 1843. Examples of Cliord algebra-valued admissible wavelets . . . . . . . . . . . . . . 188

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Cliord Algebra-valued Admissible Wavelets Associated to More

Quand sur lArbre de la Connaissanceune idee est assez mure, quelle voluptede sy insinuer, dy agir en larve,

et den precipiter la chute!

(Cioran, Syllogismes de lamertume,[11, p. 145])

Editorial Introduction

Daniel Alpay

This volume contains a selection of papers on the topics of Cliord analysis andwavelets and multiscale analysis, the latter being understood in a very wide sense.That these two topics become more and more related is illustrated for instanceby the book of Marius Mitrea [19]. The papers considering connections betweenCliord analysis and multiscale analysis constitute more than half of the presentvolume. This is maybe the specicity of this collection of articles, in comparisonwith, for instance, the volumes [12], [7], [13] or [18].

The theory of wavelets is mathematically rich and has many practical appli-cations. From a mathematical point of view it is fascinating to realize that most,if not all, of the notions arising from the theory of analytic functions in the openunit disk (in another language, the theory of discrete time systems) have coun-terparts when one replaces the integers by the nodes of a homogeneous tree. Fora review of the mathematics involved we recommand the paper of G. Letac [16].More recently, and motivated by the works of Basseville, Benveniste, Nikoukhahand Willsky (see [6], [8], [5]) the editor of this volume together with Dan Volokshowed that one can replace the complex numbers by a C-algebra built from thestructure of the tree, and dened point evaluations with values in this C-algebraand a corresponding Hardy space in which Cauchys formula holds. The pointevaluation could be used to dene in this context the counterpart of classical no-tions such as Blaschke factors. See [3], [2]. Applications include for instance theFBI ngerprint database, as explained in [15] and recalled in the introduction ofthe paper of Duktay and Jorgensen in the present volume, and the JPEG2000image compression standard.

It is also fascinating to realize that a whole function theory, dierent fromthe classical theory of several complex variables, can be developed when (say,in the quaternionic context) one considers the hypercomplex variables and theFueter polynomials and the CauchyKovalevskaya product, in place of the classicalpolynomials in three independent variables; see [10], [14]. Still, a lot of inspirationcan be drawn from the classical case, as illustrated in [1].

x D. Alpay

The volume consists of eight papers, and we now review their contents:

Classical theory: The theory of second order stationary processes indexed by thenodes of a tree involves deep tools from harmonic analysis; see [4], [9]. Some ofthese aspects are considered in the paper of H. Heyer, Stationary random eldsover graphs and related structures. The author considers in particular Karhunentype representations for stationary random elds over quotient spaces of variouskinds.

Nonlinear aspects: In the paper Teodorescu transform decomposition of multivec-tor elds on fractal hypersurfaces R. Abreu-Blaya, J. Bory-Reyes and T. Moreno-Garca consider Jordan domains with fractal boundaries. Cliord analysis toolsplay a central role in the arguments. In Methods from multiscale theory andwavelets applied to nonlinear dynamics by D. Dutkay and P. Jorgensen some newapplications of multiscale analysis are given to a nonlinear setting.

Numerical computational aspects: In the paper A Hierarchical semi-separableMoorePenrose equation solver, Patrick Dewilde and Shivkumar Chandrasekaranconsider operators with hierarchical semi-separable (HSS) structure and considertheir MoorePenrose representation. The HSS forms are close to the theory of sys-tems on trees, but here the multiresolution really represents computation states.In the paper Matrix representations and numerical computations of wavelet mul-tipliers, M.W. Wong and Hongmei Zhu use WeylHeisenberg frames to obtainmatrix representations of wavelet multipliers. Numerical examples are presented.

Connections with Cliord analysis: Such connections are studied in the paperMetric Dependent Cliord Analysis with Applications to Wavelet Analysis by F.Brackx, N. De Scheppe and F. Sommen and in the paper Cliord algebra-valuedAdmissible Wavelets Associated to more than 2-dimensional Euclidean Group withDilations by J. Zhao and L. Peng, the authors study continuous Cliord algebrawavelet transforms, and they extend to this case the classical reproducing kernelproperty of wavelet transforms; see, e.g., [17, p. 73] for the latter.

Connections with operator theory: G. Gustafson, in noncommutative trigonome-try, gives an account of noncommutative operator geometry and its applicationsto the theory of wavelets.

References

[1] D. Alpay, M. Shapiro, and D. Volok. Rational hyperholomorphic functions in R4. J.Funct. Anal., 221(1):122149, 2005.

[2] D. Alpay and D. Volok. Interpolation et espace de Hardy sur larbre dyadique: le casstationnaire. C.R. Math. Acad. Sci. Paris, 336:293298, 2003.

[3] D. Alpay and D. Volok. Point evaluation and Hardy space on a homogeneous tree.Integral Equations Operator Theory, 53:122, 2005.

[4] J.P. Arnaud. Stationary processes indexed by a homogeneous tree. Ann. Probab.,22(1):195218, 1994.

Editorial Introduction xi

[5] M. Basseville, A. Benveniste, and A. Willsky. Multiscale autoregressive processes.Rapport de Recherche 1206, INRIA, Avril 1990.

[6] M. Basseville, A. Benveniste, and A. Willsky. Multiscale statistical signal processing.In Wavelets and applications (Marseille, 1989), volume 20 of RMA Res. Notes Appl.Math., pages 354367. Masson, Paris, 1992.

[7] J. Benedetto and A. Zayed, editors. Sampling, wavelets, and tomography. Appliedand Numerical Harmonic Analysis. Birkhauser Boston Inc., Boston, MA, 2004.

[8] A. Benveniste, R. Nikoukhah, and A. Willsky. Multiscale system theory. IEEE Trans.Circuits Systems I Fund. Theory Appl., 41(1):215, 1994.

[9] W. Bloom and H. Heyer. Harmonic analysis of probability measures on hypergroups,volume 20 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin,1995.

[10] F. Brackx, R. Delanghe, and F. Sommen. Cliord analysis, volume 76. Pitman re-search notes, 1982.

[11] E.M. Cioran. Syllogismes de lamertume. Collection idees. Gallimard, 1976. Firstpublished in 1952.

[12] C.E. DAttellis and E.M. Fernandez-Berdaguer, editors. Wavelet theory and har-monic analysis in applied sciences. Applied and Numerical Harmonic Analysis.Birkhauser Boston Inc., Boston, MA, 1997. Papers from the 1st Latinamerican Con-ference on Mathematics in Industry and Medicine held in Buenos Aires, November27December 1, 1995.

[13] L. Debnath. Wavelet transforms and their applications. Birkhauser Boston Inc.,Boston, MA, 2002.

[14] R. Delanghe, F. Sommen, and V. Soucek. Cliord algebra and spinor valued func-tions, volume 53 of Mathematics and its applications. Kluwer Academic Publishers,1992.

[15] M.W. Frazier. An introduction to wavelets through linear algebra. UndergraduateTexts in Mathematics. Springer-Verlag, New York, 1999.

[16] G. Letac. Proble`mes classiques de probabilite sur un couple de Gelfand. In Analyticalmethods in probability theory (Oberwolfach, 1980), volume 861 of Lecture Notes inMath., pages 93120. Springer, Berlin, 1981.

[17] S. Mallat. Une exploration des signaux en ondelettes. Les editions de lEcole Poly-technique, 2000.

[18] Y. Meyer, editor. Wavelets and applications, volume 20 of RMA: Research Notes inApplied Mathematics, Paris, 1992. Masson.

[19] M. Mitrea. Cliord wavelets, singular integrals, and Hardy spaces, volume 1575 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1994.

Daniel AlpayDepartment of MathematicsBenGurion University of the NegevPOB 653Beer-Sheva 84105, Israele-mail: [email protected]

Operator Theory:Advances and Applications, Vol. 167, 116c 2006 Birkhauser Verlag Basel/Switzerland

Teodorescu Transform Decomposition ofMultivector Fields on Fractal Hypersurfaces

Ricardo Abreu-Blaya, Juan Bory-Reyes and Tania Moreno-Garca

Abstract. In this paper we consider Jordan domains in real Euclidean spacesof higher dimension which have fractal boundaries. The case of decomposinga Holder continuous multivector eld on the boundary of such domains isobtained in closed form as sum of two Holder continuous multivector eldsharmonically extendable to the domain and to the complement of its closurerespectively. The problem is studied making use of the Teodorescu transformand suitable extension of the multivector elds. Finally we establish equivalentcondition on a Holder continuous multivector eld on the boundary to be thetrace of a harmonic Holder continuous multivector eld on the domain.

Mathematics Subject Classication (2000). Primary: 30G35.

Keywords. Cliord analysis, Fractals, Teodorescu-transform.

1. Introduction

Cliord analysis, a function theory for Cliord algebra-valued functions in Rn

satisfying D u = 0, with D denoting the Dirac operator, was built in [10] and atthe present time becomes an independent mathematical discipline with its owngoals and tools. See [12, 18, 19] for more information and further references.

In [2, 3, 17], see also [4, 8] it is shown how Cliord analysis tools oer a lotof new brightness in studying the boundary values of harmonic dierential formsin the Hodge-de Rham sense that are in one to one correspondence with the so-called harmonic multivector elds in Rn. In particular, the case was studied ofdecomposing a Holder continuous multivector eld Fk on a closed (Ahlfors-David)regular hypersurface , which bounds a domain Rn, as a sum of two Holdercontinuous multivector elds Fk harmonically extendable to and R

n \ ( )respectively. A set of equivalent assertions was established (see [2], Theorem 4.1)which are of a pure function theoretic nature and which, in some sense, are replac-ing Condition (A) obtained by Dynkin [13] in studying the analogous problem forharmonic k-forms. For the case of C-hypersurfaces and C-multivector elds

2 R. Abreu-Blaya, J. Bory-Reyes and T. Moreno-Garca

Fk, it is revealed in [3] the equivalence between Dynkins condition (A) and theso-called conservation law ((CL)-condition)

Fk = Fk , (1)

where denotes the outer unit normal vector on and stands for the tangentialDirac operator restrictive to .

In general, a Holder continuous multivector eld Fk admits on the previ-ously mentioned harmonic decomposition if and only if

HFk = FkH, (2)

where H denotes the Hilbert transform on

Hu(x) :=2An

x y|y x|n (y)(u(y) u(x))dH

n(y) + u(x)

The commutation (2) can be viewed as an integral form of the (CL)-condition.Moreover, a Holder continuous multivector eld Fk is the trace on of a harmonicmultivector eld in if and only if HFk = Fk. An essential role in proving theabove results is played by the Cliordian Cauchy transform

Cu(x) := 1An

x y|y x|n (y)u(y)dH

n(y)

and, in particular, by the Plemelj-Sokhotski formulas, which are valid for Holdercontinuous functions on regular hypersurfaces in Rn, see [1, 6].

The question of the existence of the continuous extension of the CliordCauchy transform on a rectiable hypersurface in Rn which at the same time sat-ises the so-called AhlforsDavid regularity condition is optimally answered in [9].

What would happen with the above-mentioned decomposition for a continu-ous multivector eld if we replace the considered reasonably nice domain with onewhich has a fractal boundary?

The purpose of this paper is to study the boundary values of harmonic multi-vector elds and monogenic functions when the hypersurface is a fractal. Fractalsappear in many mathematical elds: geometric measure theory, dynamical system,partial dierential equations, etc. For example, in [25] some unexpected and in-triguing connections has been established between the inverse spectral problem forvibrating fractal strings in Rn and the famous Riemann hypothesis.

In this paper we treat mainly four kinds of problems: Let be an openbounded domain of Rn having as a boundary a fractal hypersurface .

I) (Jump Problem) Let u be a Holder continuous Cliord algebra-valued func-tion on . Under which conditions can one represent u as a sum

u = U+ + U, (3)

where the functions U are monogenic in + := and := Rn \ respec-tively?

Teodorescu Transform Decomposition 3

II) Let u be a Holder continuous Cliord algebra-valued function on . Underwhich conditions is u the trace on of a monogenic function in ?

III) (Dynkins type Problem) Let Fk be a Holder continuous multivector eld on. Under which conditions Fk can be decompose as a sum

Fk = F+k + Fk , (4)

where Fk are harmonic multivector elds in , respectively?IV) Let Fk be a Holder continuous multivector eld on . Under which conditions

is Fk the trace on of a harmonic multivector eld in ?

2. Preliminaries

We thought it to be helpful to recall some well-known, though not necessarilyfamiliar basic properties in Cliord algebras and Cliord analysis and some basicnotions about fractal dimensions and Whitney extension theorem as well.

2.1. Cliord algebras and multivectors

Let R0,n (n N) be the real vector space Rn endowed with a non-degeneratequadratic form of signature (0, n) and let (ej)nj=1 be a corresponding orthogonalbasis for R0,n. Then R0,n, the universal Cliord algebra over R0,n, is a real linearassociative algebra with identity such that the elements ej, j = 1, . . . , n, satisfythe basic multiplication rules

e2j = 1, j = 1, . . . , n;eiej + ejei = 0, i = j.

For A = {i1, . . . , ik} {1, . . . , n} with 1 i1 < i2 < < ik n, put eA =ei1ei2 eik , while for A = , e = 1 (the identity element in R0,n). Then (eA :A {1, . . . , n}) is a basis for R0,n. For 1 k n xed, the space R(k)0,n of k vectorsor k-grade multivectors in R0,n, is dened by

R(k)0,n = spanR(eA : |A| = k).

Clearly

R0,n =n

k=0

R(k)0,n.

Any element a R0,n may thus be written in a unique way asa = [a]0 + [a]1 + + [a]n

where [ ]k : R0,n R(k)0,n denotes the projection of R0,n onto Rk0,n. Notice that forany two vectors x and y, their product is given by

xy = x y + x y


where

x y = 12(xy + yx) =

nj=1

xjyj

is up to a minus sign the standard inner product between x and y, while

x y = 12(xy yx) =

i 2) and taking values in the Cliordalgebra R0,n. Such a function is said to belong to some classical class of functionsif each of its components belongs to that class.

In Rn we consider the Dirac operator:

D :=n

j=1

ejxj ,

which plays the role of the Cauchy Riemann operator in complex analysis.Due to D2 = , where being the Laplacian in Rn the monogenic functions

are harmonic.Suppose Rn is open, then a real dierentiable R0,n-valued function f

in is called left (right) monogenic in if Df = 0 (resp. fD = 0) in . Thenotion of left (right) monogenicity in Rn provides a generalization of the conceptof complex analyticity to Cliord analysis.

Many classical theorems from complex analysis could be generalized to higherdimensions by this approach. Good references are [10, 12]. The space of left mono-genic functions in is denoted by M(). An important example of a two-sidedmonogenic function is the fundamental solution of the Dirac operator, given by

e(x) =1An

x

|x|n , x Rn \ {0}.


Hereby An stands for the surface area of the unit sphere in Rn. The functione(x) plays the same role in Cliord analysis as the Cauchy kernel does in complexanalysis.

Notice that if Fk is a k-vector-valued function, i.e.,

Fk =|A|=k

eAFk,A ,

then, by using the inner and outer products (5), the action of D on Fk is given byDFk = D Fk +D Fk (6)

As DFk = Fk D with D = D and Fk = (1) k(k+1)2 Fk, it follows that if Fk is leftmonogenic, then it is right monogenic as well. Furthermore, in view of (6), Fk isleft monogenic in if and only if Fk satises in the system of equations{

D Fk = 0D Fk = 0.

(7)

A k-vector-valued function Fk satisfying (7) in is called a harmonic (k-grade)multivector eld in .

The following lemma will be much useful in what follows.

Lemma 2.1. Let u be a real dierentiable R0,n-valued function in admitting thedecomposition

u =n

k=0

[u]k.

Then u is two-sided monogenic in if and only if [u]k is a harmonic multivectoreld in , for each k = 0, 1, . . . , n.

2.3. Fractal dimensions and Whitney extension theorem

Before stating the main result of this subsection we must dene the s-dimensionalHausdor measure. Let E Rn then the Hausdor measure Hs(E) is dened by

Hs(E) := lim0

inf{

k=1

(diam Bk)s : E k Bk, diam Bk < },

where the inmum is taken over all countable -coverings {Bk} of E with open orclosed balls. Note that Hn coincides with the Lebesgue measure Ln in Rn up to apositive multiplicative constant.

Let E be a bounded set in Rn. The Hausdor dimension of E, denoted byH(E), is the inmum of the numbers s 0 such that Hs(E)


obtained from M0 by division of each of the cubes in M0 into 2nk dierent cubeswith side length 2k. Denote by Nk(E) the minimum number of cubes of the gridMk which have common points with E.

The box dimension (also called upper Minkowski dimension, see [20, 21, 22,23]) of E Rn, denoted by M(E), is dened by

M(E) = lim supk

log Nk(E)k log(2)

,

if the limit exists.The box dimension and Hausdor dimension can be equal (e.g., for the so-

called (n 1)-rectiable sets, see [16]) although this is not always valid. For setswith topological dimension n 1 we have n 1 H(E) M(E) n. If the set Ehas H(E) > n 1, then it is called a fractal set in the sense of Mandelbrot.

Let E Rn be closed. Call C0,(E), 0 < < 1, the class of R0,n-valuedfunctions satisfying on E the Holder condition with exponent .

Using the properties of the Whitney decomposition of Rn \ E (see [26], p.174) the following Whitney extension theorem is obtained.

Theorem 2.1 (Whitney Extension Theorem). Let u C0,(E), 0 < < 1. Thenthere exists a compactly supported function u C0,(Rn) satisfying(i) u|E = u|E,(ii) u C(Rn \E),(iii) |Du(x)| Cdist(x,E)1 for x Rn \E

3. Jump problem and monogenic extensions

In this section, we derive the solution of the jump problem as well as the problemof monogenically extension of a Holder continuous functions in Jordan domains.We shall work with fractal boundary data. Recall that Rn is called a Jordandomain (see [21]) if it is a bounded oriented connected open set whose boundaryis a compact topological hypersurface.

In [22, 23] Kats presented a new method for solving the jump problem, whichdoes not use contour integration and can thus be used on nonrectiable and frac-tal curves. A natural multidimensional analogue of such method was adapted im-mediately within Quaternionic Analysis in [1]. Continuing along the same lines apossible generalization for n > 2 could also be envisaged. The jump problem (3) inthis situation was considered by the authors in [6, 7]. They were able to show thatfor u C0,(), when the Holder exponent and the box dimension m := M()of the surface satisfy the relation

>mn

, (8)

then a solution of (3) can be given by the formulas

U+(x) = u(x) + TDu(x), U(x) = TDu(x)


where T is the Teodorescu transform dened by

Tv(x) =

e(y x)v(y)dLn(y).

For details regarding the basic properties of Teoderescu transform, we refer thereader to [19].

It is essential here that under condition (8) D(u|) is integrable in withany degree not exceeding nm1 (see [7]), i.e., under condition (8) it is integrablewith certain exponent exceeding n. At the same time, TDu satises the Holdercondition with exponent 1 np . Therefore we have TDu C0,(Rn) with

0 there is a constant C() such thatNk() C()2k(m+). Then,

Hn1(k) 2nC()2k(mn+1+)


As usual, the letters C,C1, C2 stand for absolute constants. Since U C0,(),U| = 0 and any point of k is distant from by no more than C1 2k, then wehave

maxyk

|U(y)| C2 2k.Consequently, for x , s = dist(x,)

|k

e(y x)(y)U(y)dHn1(y)| 2nsn1

C22kC()2k(mn+1+)

= C2C()2n

sn12k(mn+1+).

Under condition (8) the right-hand side of the above inequality tends to zero ask . By Stokes formula we have

e(y x)DU(y)dLn(y) = limk

(k

+k

)e(y x)DU(y)dLn(y)

= limk

(k

e(y x)DU(y)dLn(y)k

e(y x)(y)U(y)dHn1(y)) = 0.

ThereforeT+Du| = T+DU | = 0.

The second assertion follows directly by taking U = u+ T+Du. For the following analogous result can be obtained.

Theorem 3.2. Let u C0,(). If u is the trace of a function U C0,( ) M(), and U() = 0, then

T+Du| = u(x) (10)Conversely, if (9) is satised, then u is the trace of a function U C0,()M() and U() = 0, for some < .Remark 3.1. To be more precise, under condition (9) (resp. (10)) the functionu+ T+Du (T+Du) is a monogenic extension of u to + () which belongs toC0,(+ ) (resp. C0,( )) for any

1.d) There exist constants < 2 and C > 0 such that

|T+Du(x)| C|x|.e) There exist constants b > 0, C, such that

[T+Du(x1e1)]1 b ln(1x1

) + C, 0 < x1 1.To prove these properties we shall adapt the arguments from [22]. For the sake ofbrevity several rather technical steps will be omitted.

Proof of Theorem 5.1. Accept for the moment the validity of these properties andsuppose that (3) has a solution for the surface and function u. We then


consider the functionU(x) = u(x) + T+Du(x)

and put = U(x) (x). By c) DU Lp, p > 1, thenDU(x) = D(u(x) + T+Du(x)) = Du(x)Du(x) = 0

and U is monogenic in the domains . On the other hand, since Du(x) isbounded outside of any neighborhood of zero, then it follows that U has limitingvalues U (z) at each point of \ {0}. Moreover U+ (z) U (z) = u(z) forz \ {0} and U() = 0. Hence the function is continuous in \ {0} and itis also monogenic in R3 \ . Then, from a) and the multidimensional PainlevesTheorem proved in [1] it follows that is monogenic in R3 \ {0}. Next, by d) wehave |(x)| C|x| which implies that the singularity at the point 0 is removable(see [10]). Hence, is monogenic in R3 and () = 0. By Liouville theorem wehave (x) 0 and U(x) must be also a solution of (3) which contradicts e). 5.1. The curve of B. Kats

For the sake of clarity we rstly give a sketch of the Katss construction, which isborrowed from Reference [22], and leave the details to the reader. The constructionis as follows: Fix a number 2 (compare with [22]) and denote Mn = 2[n].Suppose {ajn}Mnj=0 are points dividing the interval In = [2n, 2n+1] into Mn equalparts: a0n = 2n+1, a1n = 2n+1 2

nMn

, etc. Denote by n the curve consisting ofthe vertical intervals [ajn, a

jn+i2

n], j = 1, . . . ,Mn1, the intervals of the real axis[a2jn , a

2j1n ], j = 1, . . . ,

Mn2 , and the horizontal intervals [a

2j+1n + i2

n, a2jn + i2n],

j = 1, . . . , Mn2 1. Then he dened =

n=1 n and constructed the closedcurve 0 consisting of the intervals [0, 1 i], [1 i, 1] and the curve . Katsshowed that M(0) = 21+ .

5.2. The surface Denote by 0 the closed plane domain bounded by the curve 0 lying on the planex3 = 0 of R3, and let 0 be the boundary of the three-dimensional closed domain

0 :={0 + e3, 12

12

}.

Next we dene the surface as the boundary of the closed cut domain

:= 0 {2x3 x1 0} {2x3 + x1 0}As it can be appreciated, the surface is composed of four plane pieces belongingto the semi-space x2 < 0 and the upper cover T := {x2 0}. Denote by Tnthe piece of T generated by the curve n. Then we have T =

n=1 Tn.

We consider the covering of T by cubes of the grid Mk. All the sets Tk+1,Tk+2, . . . , are covered by two cubes of this grid. Another two cubes cover Tk.Denote by n the distance between the vertical intervals of the curve n, i.e.n = 2

nMn

= 2n[n]. To cover the sides of Tn not more than 23k3n+1 cubes arenecessary when 2k n. When 2k < n, to cover the sides which are parallel


to the plane x1 = 0Mn1l=0

((2n+1 nl)22kn) = 22k1(3 2[n]2n + 2n)

cubes are necessary; to cover the sides which are parallel to the plane x2 = 0,not more than 22k2n+2 cubes are necessary; and to cover the sides on the planes2x3 x1 = 0 and 2x3 + x1 = 0 not more than 22k2n+1 cubes are necessary.

Therefore

4 +

2kn, k>n23k3n+1 +

2k


5.3. The function uLet 0 < < 1. Following the Katss idea we enumerate all the points ajn, j =0, . . . ,Mn, n = 1, 2, . . . in decreasing order: 0 = a01, 1 = a11, . . . . Denote byk = k k+1. Obviously, 0 = = M11 = 1, M1 = = M21 = 2,. . . , etc. Since the sequence {k}k=0 is nonincreasing, the series

j=k

(1)jj

converges. Dene the function in the following way:

(k) =j=k

(1)jj .

We extend the denition of the function to the interval [0, 1] requiring that itbe linear on all intervals [k+1, k], (0) = 0. Further we set

(x) = (x1e1 + x2e2 + x3e3) = (x1)

if x lies in the closed domain + , and (x) = 0 for x . It can be proved(see [22]) that C0,([0, 1]), then C0,(+).

Denote

= (1 + )(1 ) = 3(1 )3m , = [] 1,

32 = .

By assumption m3 we have 1, 0, and 23 < 2. Now we dene thedesired function to be u(x) = x

1(x).

5.4. Proof of properties a) e)Property a) is obvious from the construction of and the previous calculation ofits box dimension. Since x1 C0,(+) and C0,(+), then u C0,(+),u(0) = 0 and furthermore u(x) = 0 in . Hence u has property b).

On the other hand

Du(x) = e1 u(x1)x1

= e1(x11 (x1) + x

1

(x1)).

For x1 [2n, 2n+1] we have |(x1)| = 1n 2n. Hence |(x1)| 2x1 ,0 < x1 1. Furthermore (x1) n 2x11 .

From these inequalities we obtain

|Du(x)| x11 |(x1)|+ x1 |(x1)| Cx32

1 .

Since for x +, |x| 32 |x1| holds, then|Du(x)| C|x| 32.


Obviously, Du(x) Lp for p 2 . Since 23 < 2 we obtain Du(x) Lp forsome p > 1. Hence property c) follows. The same estimate proves property d). Infact,

|x||T+Du(x)| =14|+

|y x y|(x y)Du(y)|y x|3 dL

3(y)|

14|+

|y x y|Du(y)|y x|2 dL

3(y)|

C(+

|y x|2|Du(y)|dL3(y) ++

|y||Du(y)||y x|2 dL

3(y)).

Using Holder inequality the two last integrals can be bounded by a constant Cindependent of x, which proves property d). To prove e) we rstly note that

T+(Du)(x) = T+(e1(x11 (x1))(x) + T+(e1x1(x1))(x).Since (0) = 0 and C0,(+), then x11 (x) Lp(+) for some p > 3T+(e1(x11 (x1))(x) is bounded.

In order to estimate T+(e1x1(x1))(x) we split it as

T+(e1x1(x1))(x) = TP(e1x1(x1))(x) +

n=0

Tn(e1x1(x1))(x)

where

P = {x : 0 x1 1, x1 x2 0, 12x1 x3 12x1}

and

n = {x : 2n+1 x1 2n, 0 x2 h2n, 12x1 x3 12x1}

h0 = h1 = = hM12 1

= 21, hM12

= = hM22 1

= 22, . . . .

Since P has piecewise smooth boundary P and u C0,(+) , then in virtue ofthe Borel-Pompeiu formula and the Holder boundedness of the Cauchy transformCPu (see [1, 7], for instance), we have that TPDu(x) is bounded and thenTP(e1x1(x1))(x) is also bounded.

The arguments used to state the lower bound for

[

n=0

Tn(e1x1(x1))(x)]1

are rather technical and follow essentially the same Kats procedure. After that weobtain

[T+Du(x1e1)]1 C + bk=0

2k(21)

(2kx1 + 1)2, b > 0, x1 (0, 1].


Sincek=0

2k(21)

(2kx1 + 1)2

0

dt

(2tx1 + 1)2=

0

22tdt(x1 + 2t)2

ln 2 ln 1x1

then we obtain the desired inequality e).

References

[1] R. Abreu and J. Bory: Boundary value problems for quaternionic monogenic func-tions on non-smooth surfaces. Adv. Appl. Cliord Algebras, 9, No.1, 1999: 122.

[2] R. Abreu, J. Bory, R. Delanghe and F. Sommen: Harmonic multivector elds andthe Cauchy integral decomposition in Cliord analysis. BMS Simon Stevin, 11, No1, 2004: 95110.

[3] R. Abreu, J. Bory, R. Delanghe and F. Sommen: Cauchy integral decomposition ofmulti-vector-valued functions on hypersurfaces. Computational Methods and Func-tion Theory, Vol. 5, No. 1, 2005: 111134

[4] R. Abreu, J. Bory, O. Gerus and M. Shapiro: The Cliord/Cauchy transform witha continuous density; N. Davydovs theorem. Math. Meth. Appl. Sci, Vol. 28, No. 7,2005: 811825.

[5] R. Abreu, J. Bory and T. Moreno: Integration of multi-vector-valued functions overfractal hypersurfaces. In preparation.

[6] R. Abreu, D. Pena, J. Bory: Cliord Cauchy Type Integrals on Ahlfors-David RegularSurfaces in Rm+1. Adv. Appl. Cliord Algebras 13 (2003), No. 2, 133156.

[7] R. Abreu, J. Bory and D. Pena: Jump problem and removable singularities for mono-genic functions. Submitted.

[8] R. Abreu, J. Bory and M. Shapiro: The Cauchy transform for the Hodge/deRhamsystem and some of its properties. Submitted.

[9] J. Bory and R. Abreu: Cauchy transform and rectiability in Cliord Analysis. Z.Anal. Anwend, 24, No 1, 2005: 167178.

[10] F. Brackx, R. Delanghe and F. Sommen: Cliord analysis. Research Notes in Math-ematics, 76. Pitman (Advanced Publishing Program), Boston, MA, 1982. x+308 pp.ISBN: 0-273-08535-2 MR 85j:30103.

[11] F. Brackx, R. Delanghe and F. Sommen: Dierential Forms and/or MultivectorFunctions. Cubo 7 (2005), no. 2, 139169.

[12] R. Delanghe, F. Sommen and V. Soucek: Cliord algebra and spinor-valued func-tions. A function theory for the Dirac operator. Related REDUCE software by F.Brackx and D. Constales. With 1 IBM-PC oppy disk (3.5 inch). Mathematics andits Applications, 53. Kluwer Academic Publishers Group, Dordrecht, 1992. xviii+485pp. ISBN: 0-7923-0229-X MR 94d:30084.

[13] E. Dynkin: Cauchy integral decomposition for harmonic forms. Journal dAnalyseMathematique. 1997; 37: 165186.

[14] Falconer, K.J.: The geometry of fractal sets. Cambridge Tracts in Mathematics, 85.Cambridge University Press, Cambridge, 1986.


[15] Feder, Jens.: Fractals. With a foreword by Benoit B. Mandelbrot. Physics of Solidsand Liquids. Plenum Press, New York, 1988.

[16] H. Federer: Geometric measure theory. Die Grundlehren der mathematischen Wis-senschaften, Band 153, Springer-Verlag, New York Inc.,(1969) New York, xiv+676pp.

[17] J. Gilbert, J. Hogan and J. Lakey: Frame decomposition of form-valued Hardy spaces.In Cliord Algebras in analysis and related topics, Ed. J. Ryan CRC Press, (1996),Boca Raton, 239259.

[18] J. Gilbert and M. Murray: Cliord algebras and Dirac operators in harmonic anal-ysis. Cambridge Studies in Advanced Mathematics 26, Cambridge, 1991.

[19] K. Gurlebeck and W. Sprossig: Quaternionic and Cliord Calculus for Physicistsand Engineers. Wiley and Sons Publ., 1997.

[20] J. Harrison and A. Norton: Geometric integration on fractal curves in the plane.Indiana University Mathematical Journal, 4, No. 2, 1992: 567594.

[21] J. Harrison and A. Norton: The Gauss-Green theorem for fractal boundaries. DukeMathematical Journal, 67, No. 3, 1992: 575588.

[22] B.A. Kats: The Riemann problem on a closed Jordan curve. Izv. Vuz. Matematika,27, No. 4, 1983: 6880.

[23] B.A. Kats: On the solvability of the Riemann boundary value problem on a fractalarc. (Russian) Mat. Zametki 53 (1993), no. 5, 6975; translation in Math. Notes 53(1993), No. 5-6, 502505.

[24] B.A. Kats: Jump problem and integral over nonrectiable curve. Izv. Vuz. Matem-atika 31, No. 5, 1987: 4957.

[25] M.L. Lapidus and H. Maier: Hypothe`se de Riemann, cordes fractales vibrantes et con-jecture de Weyl-Berry modiee. (French) [The Riemann hypothesis, vibrating fractalstrings and the modied Weyl-Berry conjecture] C.R. Acad. Sci. Paris Ser. I Math.313 (1991), no. 1, 1924. 1991.

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Ricardo Abreu-BlayaFacultad de Informatica y MatematicaUniversidad de Holguin, Cubae-mail: [email protected]

Juan Bory-ReyesDepartamento of MatematicaUniversidad de Oriente, Cubae-mail: [email protected]

Tania Moreno-GarcaFacultad de Informatica y MatematicaUniversidad de Holguin, Cubae-mail: [email protected]

Operator Theory:Advances and Applications, Vol. 167, 1767c 2006 Birkhauser Verlag Basel/Switzerland

Metric Dependent Cliord Analysis withApplications to Wavelet Analysis

Fred Brackx, Nele De Schepper and Frank Sommen

Abstract. In earlier research multi-dimensional wavelets have been construct-ed in the framework of Cliord analysis. Cliord analysis, centered aroundthe notion of monogenic functions, may be regarded as a direct and elegantgeneralization to higher dimension of the theory of the holomorphic functionsin the complex plane. This Cliord wavelet theory might be characterized asisotropic, since the metric in the underlying space is the standard Euclideanone.

In this paper we develop the idea of a metric dependent Cliord analy-sis leading to a so-called anisotropic Cliord wavelet theory featuring waveletfunctions which are adaptable to preferential, not necessarily orthogonal, di-rections in the signals or textures to be analyzed.

Mathematics Subject Classication (2000). 42B10; 44A15; 30G35.

Keywords. Continuous Wavelet Transform; Cliord analysis; Hermite polyno-mials.

1. Introduction

During the last fty years, Cliord analysis has gradually developed to a com-prehensive theory which oers a direct, elegant and powerful generalization tohigher dimension of the theory of holomorphic functions in the complex plane.Cliord analysis focuses on so-called monogenic functions, which are in the sim-ple but useful setting of at m-dimensional Euclidean space, null solutions of theCliord-vector valued Dirac operator

=m

j=1

ejxj ,

where (e1, . . . , em) forms an orthogonal basis for the quadratic space Rm underly-ing the construction of the real Cliord algebra Rm. Numerous papers, conferenceproceedings and books have moulded this theory and shown its ability for appli-cations, let us mention [2, 15, 18, 19, 20, 24, 25, 26].

18 F. Brackx, N. De Schepper and F. Sommen

Cliord analysis is closely related to harmonic analysis in that monogenicfunctions rene the properties of harmonic functions. Note for instance that eachharmonic function h(x) can be split as h(x) = f(x)+ x g(x) with f , g monogenic,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 complex variables.The reason for this intimate relationship is that, as does the Cauchy-Riemannoperator in the complex plane, the rotation-invariant Dirac operator factorizes them-dimensional Laplace operator.

A highly important intrinsic feature of Cliord analysis is that it encompassesall dimensions at once, in other words all concepts in this multi-dimensional theoryare not merely tensor products of one dimensional phenomena but are directlydened and studied in multi-dimensional space and cannot be recursively reducedto lower dimension. This true multi-dimensional nature has allowed for amongothers a very specic and original approach to multi-dimensional wavelet theory.

Wavelet analysis is a particular time- or space-scale representation of func-tions, which has found numerous applications in mathematics, physics and engi-neering (see, e.g., [12, 14, 21]). Two of the main themes in wavelet theory arethe Continuous Wavelet Transform (abbreviated CWT) and discrete orthonormalwavelets generated by multiresolution analysis. They enjoy more or less oppositeproperties and both have their specic eld of application. The CWT plays ananalogous role as the Fourier transform and is a successful tool for the analysisof signals and feature detection in signals. The discrete wavelet transform is theanalogue of the Discrete Fourier Transform and provides a powerful technique for,e.g., data compression and signal reconstruction. It is only the CWT we are aimingat as an application of the theory developed in this paper.

Let us rst explain the idea of the one dimensional CWT. Wavelets constitutea family of functions a,b derived from one single function , called the motherwavelet, by change of scale a (i.e., by dilation) and by change of position b (i.e.,by translation):

a,b(x) =1a

(x ba

), a > 0, b R.

In wavelet theory some conditions on the mother wavelet have to be imposed.We request to be an L2-function (nite energy signal) which is well localizedboth in the time domain and in the frequency domain. Moreover it has to satisfythe so-called admissibility condition:

C := +

|(u)|2|u| du < +,

where denotes the Fourier transform of . In the case where is also in L1, thisadmissibility condition implies +

(x) dx = 0.

Metrodynamics with Applications to Wavelet Analysis 19

In other words: must be an oscillating function, which explains its qualicationas wavelet.

In practice, applications impose additional requirements, among which agiven number of vanishing moments: +

xn (x) dx = 0, n = 0, 1, . . . , N.

This means that the corresponding CWT:

F (a, b) = a,b, f

=1a

+

(x ba

)f(x) dx

will lter out polynomial behavior of the signal f up to degree N , making itadequate at detecting singularities.

The CWT may be extended to higher dimension while still enjoying the sameproperties as in the one-dimensional case. Traditionally these higher-dimensionalCWTs originate as tensor products of one-dimensional phenomena. However alsothe non-separable treatment of two-dimensional wavelets should be mentioned(see [1]).

In a series of papers [4, 5, 6, 7, 9, 10] multi-dimensional wavelets have beenconstructed in the framework of Cliord analysis. These wavelets are based onCliord generalizations of the Hermite polynomials, the Gegenbauer polynomials,the Laguerre polynomials and the Jacobi polynomials. Moreover, they arise as spe-cic applications of a general theory for constructing multi-dimensional Cliord-wavelets (see [8]). The rst step in this construction method is the introductionof new polynomials, generalizing classical orthogonal polynomials on the real lineto the Cliord analysis setting. Their construction rests upon a specic Cliordanalysis technique, the so-called Cauchy-Kowalewskaia extension of a real-analyticfunction in Rm to a monogenic function in Rm+1. One starts from a real-analyticfunction in an open connected domain in Rm, as an analogon of the classical weightfunction. The new Cliord algebra-valued polynomials are then generated by theCauchy-Kowalewskaia extension of this weight function. For these polynomialsa recurrence relation and a Rodrigues formula are established. This Rodriguesformula together with Stokess theorem lead to an orthogonality relation of thenew Cliord-polynomials. From this orthogonality relation we select candidatesfor mother wavelets and show that these candidates indeed may serve as kernelfunctions for a multi-dimensional CWT if they satisfy certain additional condi-tions.

The above sketched Cliord wavelet theory may be characterized as isotropicsince the metric in the underlying space is the standard Euclidean one for which

e2j = 1, j = 1, . . . ,mand

ejek = ekej , 1 j = k m,


leading to the standard scalar product of a vector x =m

j=1 xjej with itself:

x, x =m

j=1

x2j . (1.1)

In this paper we develop the idea of a metric dependent Cliord analysis leadingto a so-called anisotropic Cliord wavelet theory featuring wavelet functions whichare adaptable to preferential, not necessarily orthogonal, directions in the signalsor textures to be analyzed. This is achieved by considering functions taking theirvalues in a Cliord algebra which is constructed over Rm by means of a symmetricbilinear form such that the scalar product of a vector with itself now takes theform

x, x =m

j=1

mk=1

gjkxjxk. (1.2)

We refer to the tensor gjk as the metric tensor of the Cliord algebra considered,and it is assumed that this metric tensor is real, symmetric and positive denite.This idea is in fact not completely new since Cliord analysis on manifolds withlocal metric tensors was already considered in, e.g., [13], [18] and [23], while in[17] a specic three dimensional tensor leaving the third dimension unaltered wasintroduced for analyzing two dimensional signals and textures. What is new isthe detailed development of this Cliord analysis in a global metric dependentsetting, the construction of new Cliord-Hermite polynomials and the study ofthe corresponding Continuous Wavelet Transform. It should be clear that thispaper opens a new area in Cliord analysis oering a framework for a new kindof applications, in particular concerning anisotropic Cliord wavelets. We have inmind constructing specic wavelets for analyzing multi-dimensional textures orsignals which show more or less constant features in preferential directions. Forthat purpose we will have the orientation of the fundamental (e1, . . . , em)-frameadapted to these directions resulting in an associated metric tensor which willleave these directions unaltered.

The outline of the paper is as follows. We start with constructing, by meansof Grassmann generators, two bases: a covariant one (ej : j = 1, . . . ,m) anda contravariant one (ej : j = 1, . . . ,m), satisfying the general Cliord algebramultiplication rules:

ejek + ekej = 2gjk and ejek + ekej = 2gjk, 1 j, k mwith gjk the metric tensor. The above multiplication rules lead in a natural wayto the substitution for the classical scalar product (1.1) of a vector x =

mj=1 x

jejwith itself, a symmetric bilinear form expressed by (1.2). Next we generalize allnecessary denitions and results of orthogonal Cliord analysis to this metric de-pendent setting. In this new context we introduce for, e.g., the concepts of Fischerinner product, Fischer duality, monogenicity and spherical monogenics. Similar tothe orthogonal case, we can also in the metric dependent Cliord analysis decom-pose each homogeneous polynomial into spherical monogenics, which is referred


to as the monogenic decomposition. After a thorough investigation in the met-ric dependent context of the so-called Euler and angular Dirac operators, whichconstitute two fundamental operators in Cliord analysis, we proceed with the in-troduction of the notions of harmonicity and spherical harmonics. Furthermore, weverify the orthogonal decomposition of homogeneous polynomials into harmonicones, the so-called harmonic decomposition. We end Section 2 with the denitionand study of the so-called g-Fourier transform, the metric dependent analogue ofthe classical Fourier transform.

With a view to integration on hypersurfaces in the metric dependent setting,we invoke the theory of dierential forms. In Subsection 3.1 we gather some basicdenitions and properties concerning Cliord dierential forms. We discuss, e.g.,fundamental operators such as the exterior derivative and the basic contractionoperators; we also state Stokess theorem, a really fundamental result in mathe-matical analysis. Special attention is paid to the properties of the Leray and sigmadierential forms, since they both play a crucial role in establishing orthogonalityrelations between spherical monogenics on the unit sphere, the topic of Subsec-tion 3.2.

In a fourth and nal section we construct the so-called radial g-Cliord-Hermite polynomials, the metric dependent analogue of the radial Cliord-Hermitepolynomials of orthogonal Cliord analysis. For these polynomials a recurrenceand orthogonality relation are established. Furthermore, these polynomials turnout to be the desired building blocks for specic wavelet kernel functions, the so-called g-Cliord-Hermite wavelets. We end the paper with the introduction of thecorresponding g-Cliord-Hermite Continuous Wavelet Transform.

2. The metric dependent Cliord toolbox

2.1. Tensors

Let us start by recalling a few concepts concerning tensors. Assume in Euclideanspace two coordinate systems (x1, x2, . . . , xN ) and (x1, x2, . . . , xN ) given.

Denition 2.1. If (A1, A2, . . . , AN ) in coordinate system (x1, x2, . . . , xN ) and(A1, A2, . . . , AN ) in coordinate system (x1, x2, . . . , xN ) are related by the trans-formation equations:

Aj =N

k=1

xj

xkAk , j = 1, . . . , N,

then they are said to be components of a contravariant vector.


Denition 2.2. If (A1, A2, . . . , AN ) in coordinate system (x1, x2, . . . , xN ) and(A1, A2, . . . , AN ) in coordinate system (x1, x2, . . . , xN ) are related by the trans-formation equations:

Aj =N

k=1

xk

xjAk, j = 1, . . . , N,

then they are said to be components of a covariant vector.

Example. The sets of dierentials {dx1, . . . , dxN} and {dx1, . . . , dxN} transformaccording to the chain rule:

dxj =N

k=1

xj

xkdxk, j = 1, . . . , N.

Hence (dx1, . . . , dxN ) is a contravariant vector.

Example. Consider the coordinate transformation(x1, x2, . . . , xN

)=(x1, x2, . . . , xN

)A

with A = (ajk) an (N N)-matrix. We have

xj =N

k=1

xkajk or equivalently xj =

Nk=1

xj

xkxk,

which implies that (x1, . . . , xN ) is a contravariant vector.

Denition 2.3. The outer tensorial product of two vectors is a tensor of rank 2.There are three possibilities: the outer product of two contravariant vectors (A1,...,AN ) and (B1,...,BN)

is a contravariant tensor of rank 2:

Cjk = AjBk

the outer product of a covariant vector (A1, . . . , AN ) and a contravariantvector (B1, . . . , BN ) is a mixed tensor of rank 2:

Ckj = AjBk

the outer product of two covariant vectors (A1, . . . , AN ) and (B1, . . . , BN ) isa covariant tensor of rank 2:

Cjk = AjBk.

Example. The Kronecker-delta

kj =

{1 if j = k0 otherwise

is a mixed tensor of rank 2.


Example. The transformation matrix

ajk =xj

xk, j, k = 1, . . . , N

is also a mixed tensor of rank 2.

Remark 2.4. In view of Denitions 2.1 and 2.2 it is easily seen that the transfor-mation formulae for tensors of rank 2 take the following form:

Cjk = AjBk =Ni=1

xj

xiAi

N=1

xk

xB

=Ni=1

N=1

xj

xixk

xAiB =

Ni=1

N=1

xj

xixk

xCi

and similarly

Ckj =Ni=1

N=1

xi

xjxk

xCi , Cjk =

Ni=1

N=1

xi

xjx

xkCi.

Denition 2.5. The tensorial contraction of a tensor of rank p is a tensor of rank(p2) which one obtains by summation over a common contravariant and covariantindex.

Example. The tensorial contraction of the mixed tensor Cjk of rank 2 is a tensorof rank 0, i.e., a scalar:

Nj=1

Cjj = D,

while the tensorial contraction of the mixed tensor Cjki of rank 3 yields a con-travariant vector:

Nj=1

Cjkj = Dk.

Denition 2.6. The inner tensorial product of two vectors is their outer productfollowed by contraction.

Example. The inner tensorial product of the covariant vector (A1, . . . , AN ) and thecontravariant vector (B1, . . . , BN ) is the tensor of rank 0, i.e., the scalar given by:

Nj=1

AjBj = C.


2.2. From Grassmann to Cliord

We consider the Grassmann algebra generated by the basis elements (j , j =1, . . . ,m) satisfying the relations

j k + k j = 0 1 j, k m.These basis elements form a covariant tensor (1, 2, . . . , m) of rank 1.

Next we consider the dual Grassmann algebra + generated by the dualbasis (+j , j = 1, . . . ,m), forming a contravariant tensor of rank 1 and satisfyingthe Grassmann identities:

+j

+k + +k +j = 0 1 j, k m. (2.1)Duality between both Grassmann algebras is expressed by:

j +k + +k j =

kj . (2.2)

Note that both the left- and right-hand side of the above equation is a mixedtensor of rank 2.

Now we introduce the fundamental covariant tensor gjk of rank 2. It is as-sumed to have real entries, to be positive denite and symmetric:

gjk = gkj , 1 j, k m.Denition 2.7. The real, positive denite and symmetric tensor gjk is called themetric tensor.

Its reciprocal tensor (a contravariant one) is given by

gjk =1

det(gjk)Gjk,

where Gjk denotes the cofactor of gjk. It thus satisesm=1

gjgk = jk.

In what follows we will use the Einstein summation convention, i.e., summationover equal contravariant and covariant indices is tacitly understood.

With this convention the above equation expressing reciprocity is written as

gjgk = jk.

Denition 2.8. The covariant basis (+j , j = 1, . . . ,m) for the Grassmann algebra+ is given by

+j = gjk

+k.

This covariant basis shows the following properties.

Proposition 2.9. One has

+j

+k +

+k

+j = 0 and j

+k +

+k j = gjk ; 1 j, k m.


Proof. By means of respectively (2.1) and (2.2), we nd

+j

+k +

+k

+j = gj

+gkt+t + gkt+tgj+

= gjgkt(++t + +t+) = 0

and

j +k +

+k j = jgkt

+t + gkt+tj = gkt(j +t + +tj)

= gkttj = gkj = gjk. Remark 2.10. By reciprocity one has:

+k = gkj +j .

Denition 2.11. The contravariant basis (j , j = 1, . . . ,m) for the Grassmannalgebra is given by

j = gjkk.

It shows the following properties.


j

k + k j = 0 and j +k + +k j = gjk.

Proof. A straightforward computation yields

j

k + k j = gjgkt

t + gkt

tgj

= gjgkt(t + t) = 0

and

j

+k + +kj = gj+k + +kgj

= gj(+k + +k )

= gjk = gjk.

Remark 2.13. By reciprocity one has:

k = gkj j .

Now we consider in the direct sum

spanC{1, . . . , m} spanC{+1 , . . . , +m}two subspaces, viz. spanC{e1, . . . , em} and spanC{em+1, . . . , e2m} where the newcovariant basis elements are dened by{

ej = j +j , j = 1, . . . ,mem+j = i(j +

+j ), j = 1, . . . ,m.

Similarly, we consider the contravariant reciprocal subspaces spanC{e1, . . . , em}and spanC{em+1, . . . , e2m} given by{

ej = j +j , j = 1, . . . ,mem+j = i(j + +j), j = 1, . . . ,m.


The covariant basis shows the following properties.

Proposition 2.14. For 1 j, k m one has:(i) ejek + ekej = 2gjk(ii) em+jem+k + em+kem+j = 2gjk(iii) ejem+k + em+kej = 0.

Proof. A straightforward computation leads to

(i) ejek + ekej = (j +j )(k +k ) + (k +k )(j +j )= j k j +k +j k + +j +k + kj k +j +k j + +k +j= (j k + k j) + (

+j

+k +

+k

+j ) (j +k + +k j) (+j k + k +j )

= gjk gkj = 2gjk,(ii) em+jem+k + em+kem+j

= (j + +j )(k + +k ) (k + +k )(j + +j )= j k j +k +j k +j +k k j k +j +k j +k +j= (j +k + +k j) (+j k + k +j )= gjk gkj = 2gjk

and(iii) ejem+k + em+kej

= i(j +j )(k + +k ) + i(k + +k )(j +j )= ij k + ij

+k i+j k i+j +k + ik j ik +j + i+k j i+k +j

= i(j k + k j) i(+j +k + +k +j ) + i(j +k + +k j) i(+j k + k +j )= igjk igkj = 0.

As expected, both e-bases are linked to each other by means of the metrictensor gjk and its reciprocal gjk.

Proposition 2.15. For j = 1, . . . ,m one has(i) ej = gjkek and ek = gkjej

(ii) em+j = gjkem+k and em+k = gkjem+j.

Proof.

(i) ej = j +j = gjkk gjk+k= gjk(k +k ) = gjkek

(ii) em+j = i(j + +j) = igjk(k + +k )

= gjkem+k.

By combining Propositions 2.14 and 2.15 we obtain the following propertiesof the contravariant e-basis.


Proposition 2.16. For 1 j, k m one has(i) ejek + ekej = 2gjk(ii) em+jem+k + em+kem+j = 2gjk(iii) ejem+k + em+kej = 0.

The basis (e1, . . . , em) and the dual basis (e1, . . . , em) are also linked to eachother by the following relations.

Proposition 2.17. For 1 j, k m one has(i) ejek + ekej = 2kj(ii) em+jem+k + em+kem+j = 2kj(iii) ejem+k + em+kej = 0(iv)

j eje

j =

j ejej = m

(v) 12

j{ej, ej} = m(vi)

j [ej , e

j] = 0(vii)

j em+je

m+j =

j em+jem+j = m

(viii) 12

j{em+j, em+j} = m(ix)

j [em+j , e

m+j] = 0.


(i) ejek + ekej = ejgktet + gktetej = gkt(ejet + etej)

= gkt(2gjt) = 2kj(ii) em+jem+k + em+kem+j = em+jgktem+t + gktem+tem+j

= gkt(em+jem+t + em+tem+j)

= gkt(2gjt) = 2kj(iii) ejem+k + em+kej = ejgktem+t + gktem+tej

= gkt(ejem+t + em+tej) = 0

(iv)m

j=1

ejej =

mj=1

ej

mi=1

gjiei =i,j

gijejei

=12

i,j

gijejei +12

i,j

gijejei =12

i,j

gijejei +12

j,i

gjieiej

=12

i,j

gijejei +12

i,j

gij(ejei 2gij) = i,j

gijgij

= i

ii = (1 + . . .+ 1) = m.

By means of (i) we also havem

j=1

ejej =m

j=1

(ejej 2) = m 2m = m.


(v) Follows directly from (iv).(vi) Follows directly from (iv).(vii) Similar to (iv).(viii) Follows directly from (vii).(ix) Follows directly from (vii).

Finally we consider the algebra generated by either the covariant basis (ej :j = 1, . . . ,m) or the contravariant basis (ej : j = 1, . . . ,m) and we observe thatthe elements of both bases satisfy the multiplication rules of the complex Cliordalgebra Cm:

ejek + ekej = 2gjk, 1 j, k mand

ejek + ekej = 2gjk, 1 j, k m.A covariant basis for Cm consists of the elements eA = ei1ei2 . . . eih where A =(i1, i2, . . . , ih) {1, . . . ,m} = M is such that 1 i1 < i2 < . . . < ih m.Similarly, a contravariant basis for Cm consists of the elements eA = ei1ei2 . . . eihwhere again A = (i1, i2, . . . , ih) M is such that 1 i1 < i2 < . . . < ih m. Inboth cases, taking A = , yields the identity element, i.e., e = e = 1.

Hence, any element Cm may be written as =

A

AeA or as =A

AeA with A C.

In particular, the space spanned by the covariant basis (ej : j = 1, . . . ,m) or thecontravariant basis (ej : j = 1, . . . ,m) is called the subspace of Cliord-vectors.

Remark 2.18. Note that the real Cliord-vector = jej may be considered asthe inner tensorial product of a contravariant vector j with real elements witha covariant vector ej with Cliord numbers as elements, which yields a tensor ofrank 0. So the Cliord-vector is a tensor of rank 0; in fact it is a Cliord number.

In the Cliord algebra Cm, we will frequently use the anti-involution calledHermitian conjugation, dened by

ej = ej, j = 1, . . . ,mand

=(

A

AeA) =

A

cAeA,

where cA denotes the complex conjugate of A. Note that in particular for a realCliord-vector = jej : = .

The Hermitian inner product on Cm is dened by

(, ) = []0 C,where []0 denotes the scalar part of the Cliord number . It follows that for Cm its Cliord norm ||0 is given by

||20 = (, ) = []0.


Finally let us examine the Cliord product of the two Cliord-vectors = jej and = jej:

= jejkek = jkejek

=12jkejek +

12jk(ekej 2gjk)

= gjkjk + 12jk(ejek ekej).

It is found that this product splits up into a scalar part and a so-called bivectorpart:

= + with

= gjkjk = 12( + ) =12{, }

the so-called inner product and

= 12jk(ejek ekej) = 12

jk[ej, ek] =12[, ]

the so-called outer product.In particular we have that

ej ek = gjk, 1 j, k mej ej = 0, j = 1, . . . ,m

and similarly for the contravariant basis elements:

ej ek = gjk, 1 j, k mej ej = 0, j = 1, . . . ,m.

The outer product of k dierent basis vectors is dened recursively

ei1 ei2 eik =12(ei1(ei2 eik) + (1)k1(ei2 eik)ei1

).

For k = 0, 1, . . . ,m xed, we then call

Ckm =

{ Cm : =

|A|=k

A ei1 ei2 eik , A = (i1, i2, . . . , ik)}

the subspace of k-vectors ; i.e., the space spanned by the outer products of k dif-ferent basis vectors. Note that the 0-vectors and 1-vectors are simply the scalarsand Cliord-vectors; the 2-vectors are also called bivectors.


2.3. Embeddings of Rm

By identifying the point (x1, . . . , xm) Rm with the 1-vector x given byx = xjej ,

the space Rm is embedded in the Cliord algebra Cm as the subspace of 1-vectors R1m of the real Cliord algebra Rm. In the same order of ideas, a point(x0, x1, . . . , xm) Rm+1 is identied with a paravector x = x0 + x in R0m R1m.

We will equip Rm with a metric by dening the scalar product of two basisvectors through their dot product:

ej , ek = ej ek = gjk, 1 j, k mand by putting for two vectors x and y:

x, y = ejxj , ekyk = ej , ekxjyk = gjkxjyk.Note that in this way . , . is indeed a scalar product, since the tensor gjk issymmetric and positive denite. Also note that in particular

x2 = x, x = gjkxjxk = x x = x2 = |x|20 .We also introduce spherical coordinates in Rm by:

x = r

withr = x = (gjkxjxk)1/2 [0,+[ and Sm1,

where Sm1 denotes the unit sphere in Rm:

Sm1 ={ R1m ; 2 = 2 = gjkjk = 1

}.

Now we introduce a new basis for Rm = R1m consisting of eigenvectors of thematrix G = (gjk), associated to the metric tensor gjk. As (gjk) is real-symmetric,there exists an orthogonal matrix A O(m) such that

ATGA = diag(1, . . . , m)

with 1, . . . , m the positive eigenvalues of G. We put

(Ej) = (ej)A.

We expect the basis (Ej : j = 1, . . . ,m) to be orthogonal, since Ej (j = 1, . . . ,m)are eigenvectors of the matrix G.

If (xj) and (Xj) are the column matrices representing the coordinates of xwith respect to the bases (ej) and (Ej) respectively, then we have

x = (ej)(xj) = (Ej)(Xj) = (ej)A(Xj)

and(xj) = A(Xj) or (Xj) = AT (xj).


Hence

x, y = (xj)TG(yj) = (Xj)TATGA(Y j)= (Xj)T diag(1, . . . , m) (Y j) =

j

jXjY j .

In particular we nd, as expected:

Ej , Ek = 0, j = k and Ej , Ej = j , j = 1, . . . ,m.Involving the Cliord product we obtain

{Ej, Ek} = 2Ej Ek = 2Ej, Ek = 0, j = kand

{Ej , Ej} = 2Ej Ej = 2Ej , Ej = 2j, j = 1, . . . ,mand so for all 1 j, k m

EjEk + EkEj = 2jjk.Finally if we put

j=

Ejj

(j = 1, . . . ,m),

we obtain an orthonormal frame in the metric dependent setting:

j,

k = jk; 1 j, k m.

In what follows, (xj) denotes the column matrix containing the coordinates of xwith respect to this orthonormal frame, i.e.,

x =j

xj j=j

j, x

j.

It is clear that the coordinate sets (Xj) and (xj) are related as follows:

(Xj) = P1(xj) or (xj) = P (Xj)

withP = diag(

1, . . . ,

m).

Hence we also have

(xj) = AP1(xj) and (xj) = PAT (xj).

Finally, note that in the xj -coordinates, the inner product takes the followingform

x, y =j

xjyj .


2.4. Fischer duality and Fischer decomposition

We consider the algebra P of Cliord algebra-valued polynomials, generated by{x1, . . . , xm ; e1, . . . , em; i}. A natural inner product on P is the so-called Fischerinner product(

R(x), S(x))=[{

R(g1jxj , g

2jxj , . . . , gmjxj

)[S(x)]

}x=0

]0

.

Note that in order to obtain the dierential operator R(g1jxj , . . . , gmjxj ), onerst takes the Hermitian conjugate R of the polynomial R, followed by the sub-stitution xk gkjxj . These two operations{

ekF ek (Hermitian conjugation)

xkF gkjxj

are known as Fischer duality.Now we express this Fischer duality in terms of the new basis (Ej : j =

1, . . . ,m) and the corresponding new coordinates (Xj) introduced in the foregoingsubsection:

(i) (Ej) = (ej)AF (ej)A = (ej)A = (Ej) (Hermitian conjugation)

(ii) Xj = ATjkxk F ATjkgikxi = ATjkgik

X

xiX = A

Tjkg

ikATiX

= ATjkgikAi X = (A

TG1A)j X

=(diag

(11

, . . . ,1m

))j

X =1j

Xj . (2.3)

Proposition 2.19. The basis(EAX

= EA(X1)1 . . . (Xm)m : A M, Nm)

of the space P of Cliord polynomials is orthogonal with respect to the Fischerinner product.

Proof. We have consecutively(EAX

, EBX)

=[EA

(11

X1

)1. . .

(1m

Xm

)m[EBX ]

]0

/X=0

=[EAEB

(11

)1. . .

(1m

)m(X1)

1 . . . (Xm)m (X1)1 . . . (Xm)m

]0

/X=0

=(

11

)1. . .

(1m

)m! , [E

AEB ]0

with ! = 1! . . . m! . Moreover, for

EA = Ei1Ei2 . . . Eih and EB = Ej1Ej2 . . . Ejk ,

we ndEAEB = (1)|A| Eih . . . Ei2Ei1Ej1Ej2 . . . Ejk .


AsEjEk + EkEj = 2jjk, 1 j, k m,

we have

[EAEB]0 = (1)|A| (i1)(i2 ) . . . (ih) A,B= i1i2 . . . ih A,B.

Summarizing, we have found that for A = (i1, i2, . . . , ih)

(EAX, EBX) = ! i1i2 . . . ih

(11

)1. . .

(1m

)mA,B , .

Proposition 2.20. The Fischer inner product is positive denite.

Proof. This follows in a straightforward way from the fact that the Fischer innerproduct of a basis polynomial EAX with itself is always positive:(

EAX, EAX

)= ! i1i2 . . . ih

(11

)1. . .

(1m

)m> 0.

Let Pk denote the subspace of P consisting of the homogeneous Cliordpolynomials of degree k:

Pk = {Rk(x) P : Rk(tx) = tkRk(x) , t R}.It follows from Proposition 2.19 that the spaces Pk are orthogonal with respectto the Fischer inner product. With a view to the Fischer decomposition of thehomogeneous Cliord polynomials, we now introduce the notion of monogenicity,which in fact is at the heart of Cliord analysis in the same way as the notionof holomorphicity is fundamental to the function theory in the complex plane.Monogenicity is dened by means of the so-called Dirac-operator x which weintroduce as the Fischer dual of the vector variable x:

x = xjejF gjkxkej = ekxk = x .

This Dirac operator factorizes the g-Laplacian, which we obtain as the Fischerdual of the scalar function x2 = x2:x2=x,x=gjkxjxk Fgjkgjixigkx =gjkgjigkxix =ikgkxix

=gi2xix =g,where we have dened the g-Laplacian as to be

g = gjk2xjxk .

Then we have indeed that

2x = xjejxke

k =12xjxke

jek +12xkxje

kej

=12xjxke

jek +12xjxk(ejek 2gjk)

=12xjxke

jek 12xjxke

jek gjkxjxk = g.


We also mention the expression for the Dirac operator in the orthonormal frameintroduced at the end of Subsection 2.3.

Lemma 2.21. With respect to the orthonormal frame j(j = 1, . . . ,m), the Dirac

operator takes the form

x =j

xjj .

Proof. For the sake of clarity, we do not use the Einstein summation convention.We have consecutively

x =j

xjej =

j,k

xjgjkek =

j,k,t

xjgjkEtA

Ttk

=t

j,k

ATtkgjkxj

Et.By means of (2.3) this becomes

x =t

1t

XtEt =t

xtt.

Denition 2.22.

(i) A Cm-valued function F (x1, . . . , xm) is called left monogenic in an open re-gion of Rm if in that region

x[F ] = 0.

(ii) A Cm-valued function F (x0, x1, . . . , xm) is called left monogenic in an openregion of Rm+1 if in that region

(x0 + x)[F ] = 0.

Here x0 + x is the so-called Cauchy-Riemann operator.The notion of right monogenicity is dened in a similar way by letting act

the Dirac operator or the Cauchy-Riemann operator from the right.Note that if a Cliord algebra-valued function F is left monogenic, then its

Hermitian conjugate F is right monogenic, since x = x.Similar to the notion of spherical harmonic with respect to the Laplace op-

erator we introduce the following fundamental concept.

Denition 2.23.

(i) A (left/right) monogenic homogeneous polynomial Pk Pk is called a(left/right) solid inner spherical monogenic of order k.

(ii) A (left/right) monogenic homogeneous function Qk of degree (k + m 1)in Rm \ {0} is called a (left/right) solid outer spherical monogenic of order k.


The set of all left, respectively right, solid inner spherical monogenics of order k willbe denoted by M+ (k), respectively M

+r (k), while the set of all left, respectively

right, solid outer spherical monogenics of order k will be denoted by M (k),respectively Mr (k).

Theorem 2.24 (Fischer decomposition).

(i) Any Rk Pk has a unique orthogonal decomposition of the formRk(x) = Pk(x) + xRk1(x)

with Pk M+ (k) and Rk1 Pk1.(ii) The space of homogeneous polynomials Pk admits the orthogonal decomposi-

tion: Pk = M+ (k) xPk1.Proof. This orthogonal decomposition follows from the observation that forRk1 Pk1 and Sk Pk:

(xRk1(x), Sk(x)) = (Rk1(x), x[Sk(x)]). (2.4)Indeed, as x is the Fischer dual of x, we have

(xRk1(x), Sk(x)) = [Rk1

(g1jxj , . . . , g

mjxj)x[Sk(x)]

]0

= (Rk1(x), x[Sk(x)]) .Next, if for some Sk Pk and for all Rk1 Pk1

(xRk1, Sk) = 0,

then so will(Rk1, x[Sk]) = 0.

Hence x[Sk] = 0, which means that the orthogonal complement of xPk1 is asubspace of M+ (k) .

But if Pk = xRk1 M+ (k) xPk1 , then(Pk, Pk) = (xRk1, Pk) = (Rk1, x[Pk]) = 0

and thus Pk = 0. So any Sk Pk may be uniquely decomposed asSk = Pk + xRk1 with Pk M+ (k) and Rk1 Pk1.

Theorem 2.25 (monogenic decomposition). Any Rk Pk has a unique orthogonaldecomposition of the form

Rk(x) =k

s=0

xsPks(x), with Pks M+ (k s).

Proof. This result follows by recursive application of the Fischer decomposition.


2.5. The Euler and angular Dirac operators

The Euler and angular Dirac operators are two fundamental operators arising quitenaturally when considering the operator xx; in fact they are the respective scalarand bivector part of it.

Denition 2.26. The Euler operator is the operator dened by

E = xjxj ;

the angular Dirac operator is dened by

= 12xjxk(eje

k ekej) = xjxkej ek.Proposition 2.27. One has

xx = E (2.5)or in other words

E = x x = [xx]0 and = x x = [xx]2,where []2 denotes the bivector part of the Cliord number .

Proof. One easily nds

xx = xjejxkek =

12xjxkeje

k +12xjxkeje

k

=12xjxkeje

k +12xjxk(ekej 2kj )

= xjxj + 12xjxk(eje

k ekej) = E . As is well known, the Euler operator measures the degree of homogeneity of poly-nomials, while the angular Dirac operator measures the degree of monogenicity.This is expressed by the following eigenvalue equations.

Proposition 2.28.

(i) For Rk Pk one hasE[Rk] = kRk.

(ii) For Pk M+ (k) one has[Pk] = kPk.

Proof. (i) A homogeneous polynomial Rk of degree k can be written as

Rk(x) =A

eA Rk,A(x)

with Rk,A a scalar-valued homogeneous polynomial of degree k. Hence

E[Rk] =A

eA E[Rk,A] =A

eAkRk,A = kRk.

(ii) Using (2.5) it is easily seen that

[Pk] = (E xx)[Pk] = kPk.


Remark 2.29. A Cliord polynomial operator is an element of End(P); it trans-forms a Cliord polynomial into another one. Such a Cliord polynomial operatorA is called homogeneous of degree if

A[Pk] Pk+.The Euler operator also measures the degree of homogeneity of Cliord polynomialoperators.

Proposition 2.30. The Cliord polynomial operator A is homogeneous of degree if and only if

[E,A] = A.

Proof. Suppose that A is homogeneous of degree . We then have

(EA AE)[Pk] = E[Pk+] kA[Pk] = (k + )Pk+ kPk+ = A[Pk].Conversely, assume that the Cliord polynomial operator B satises

[E,B] = B.

For an arbitrary Rk Pk we then haveE(B[Rk]) = (BE + B)[Rk] = kB[Rk] + B[Rk] = (k + )B[Rk].

In other words, B[Rk] Pk+ and hence B is homogeneous of degree . In a series of lemmata and propositions we now establish the basic formulae

and operator identities needed in the sequel.

Lemma 2.31. One hasx[x] = m.

Proof. By means of Proposition 2.17, we have immediately

x[x] =k,j

ekxk(xj)ej =

j

ejej = m.


xx + xx = 2E m and xx = E m + . (2.6)Proof. On the one hand we have

xx =j,k

xjejxkek,

while on the other hand

xx = x[x] + xx = m+j,k

xkekxjej = m+

j,k

xjxkekej ,

where the dot-notation x means that the Dirac operator does not act on thefunction x, but on the function at the right of it.


Adding both equalities yields

xx + xx = m+j,k

xjxk(ejek + ekej)

= m 2j,k

xjxkkj = m 2E.

Using (2.5) this becomes

xx = xx 2E m = E + 2E m = E m.

Propositions 2.27 and 2.32 yield some nice additional results. The rst corollaryfocuses on an interesting factorization of the g-Laplacian.

Corollary 2.33. One has

g = (E +m ) 1||x||2 (E + ).

Proof. By means of (2.5) and (2.6), we obtain

g = 2x = xx2

||x||2 x = x x1

||x||2 x x

= (E +m ) 1||x||2 (E + ).

Also the polynomials xPk are eigenfunctions of the angular Dirac operator.

Corollary 2.34. For any Pk M+ (k) one has:[xPk] = (k +m 1)xPk.

Proof. By means of (2.6) we obtain

xx[xPk] = x(E m+ )[Pk]= x(k m k)Pk = (2k m)xPk.

Using (2.5) and the fact that xPk Pk+1 gives(2k m)xPk = (E + )[xPk] = (k + 1)xPk [xPk].

Hence[xPk] = (k +m 1)xPk.

Lemma 2.35. One has(i) xj [r] =

1rgjkx

k , j = 1, . . . ,m

(ii) x[r] =x

r.



(i) xj [r] = xj[(

gikxixk)1/2] = 1

2(gikx

ixk)1/2(

gjkxk + gijxi

)=

121r2gjkxk =

1rgjkx

k

(ii) x[r] = ejxj [r] =1rejgjkx

k =1rekx

k =x

r.

Now it is easily proved that the angular Dirac operator only acts on the angularcoordinates, whence its name.

Lemma 2.36. One has(i) E[r] = r(ii) xx[r] = r(iii) [r] = 0.

Proof. By means of the previous Lemma we easily nd

(i) E[r] = xjxj [r] = xj1rgjkx

k =1rr2 = r.

(ii) xx[r] = xx

r= r

2

r= r.

(iii) [r] = (xx + E)[r] = (r + r) = 0. This enables us to prove the following eigenvalue equations.

Theorem 2.37. One has(i) E[xsPk] = (s + k)xsPk(ii) [x2sPk] = kx2sPk(iii) [x2s+1Pk] = (k + m 1)x2s+1Pk.Proof. (i) Follows immediately from the fact that xsPk Ps+k .(ii) By means of the previous Lemma we nd

[x2sPk] = x2s[Pk] = kx2sPk.(iii) Similarly we have

[x2s+1Pk] = x2s[xPk] = (k +m 1)x2s+1Pk. The previous Theorem combined with (2.6) yields the following result.

Theorem 2.38. One hasx[xsPk] = Bs,k xs1Pk

with

Bs,k =

{s for s even,(s 1 + 2k +m) for s odd.


Proof. We have consecutively

x[x2sPk] = xx[x2s1Pk] = (E m+ )[x2s1Pk]= (2s+ k 1)x2s1Pk mx2s1Pk + (k +m 1)x2s1Pk= 2sx2s1Pk

and similarly

x[x2s+1Pk] = xx[x2sPk] = (E m+ )[x2sPk]= (2s+ k)x2sPk mx2sPk kx2sPk= (2s+ 2k +m)x2sPk.

Solid inner and outer spherical monogenics are related to each other by meansof the so-called spherical transform.

Proposition 2.39. If Pk is a solid inner spherical monogenic of degree k, then

Qk(x) =x

xm Pk(

x

x2)

=x

x2k+mPk(x)

is a monogenic homogeneous function of degree (k +m 1) in Rm \ {0}, i.e., asolid outer spherical monogenic of degree k.

Conversely, if Qk is a solid outer spherical monogenic of degree k, then

Pk(x) =x

xm Qk(

x

x2)

= x x2k+m2 Qk(x)

is a solid inner spherical monogenic of degree k.

Proof. Clearly Qk is homogeneous of degree (k +m 1), since

Qk(tx) =tk+1xPk(x)

t2k+m||x||2k+m = t(k+m1)Qk(x).

Moreover, in Rm \ {0}:

x[Qk(x)] = x

[1

||x||2k+m]xPk +

1||x||2k+m x[xPk]

= (2k +m) x||x||2k+m+2 xPk (2k +m)1

||x||2k+mPk

=(2k +m)||x||2k+m Pk

(2k +m)||x||2k+m Pk = 0,

where we have used

x

[1

r2k+m

]= (2k +m) 1

r2k+m+1x[r] = (2k +m)

r2k+m+1x

r= (2k +m) x

r2k+m+2.

The converse result is proved in a similar way.


Proposition 2.40. In terms of spherical coordinates the Euler operator E takes theform

E = rr .

Proof. In this proof we do not use the Einstein summation convention.The transformation formulae from cartesian to spherical coordinates in Rm

yield

xj =r

xjr +

mk=1

k

xjk . (2.7)

In view of Lemma 2.35 we have for each j xed:r

xj=

1r

t

gjtxt,

andj

xj=

xj

(xj

r

)=

1r xj 1

r2r

xj

=1r 1

r3xjt

gjtxt =

1r

(1 j

t

gjtt),

while for k = j, we ndk

xj=

xj

(xk

r

)= xk 1

r2r

xj

= xk 1r3

t

gjtxt = k 1

r

t

gjtt.

Hence equation (2.7) becomes

xj =1r

(t

gjtxt)r +

1r

(1 j(

t

gjtt))j

mk=1, k =j

k1r(t

gjtt)k

=1r

(t

gjtxt)r +

1rj 1r

mk=1

k(t

gjtt)k . (2.8)

Thus we nally obtain

E=j

xjxj =1r

j

xj(

t

gjtxt)r+

1r

j

xjj 1r

j

xjk

k(t

gjtt)k

=1r

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