Radon type transforms in Clifford analysis

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Radon type transformaties in cliffordanalyse

Radon type transforms in Clifford analysis

Ren Hu

Promotoren:

Prof. Dr. Hendrik De Bie Dr. Tim Raeymaekers Dr. Al´ı Guzm´an Ad´an

Proefschrift voorgelegd aan de Faculteit Wetenschappen van de Universiteit Gent tot het behalen van de graad van doctor in de wetenschappen: wiskunde.

Universiteit Gent

Faculteit Ingenieurswetenschappen en Architectuur

Vakgroep Elektronica en Informaticasystemen - Onderzoekseenheid Cliffordanalyse

Academiejaar 2020-2021

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Acknowledgements

In general, four years would be a very long time. But my first day in Gent after a four-year study is still like yesterday. Till this moment, I remember the first time I was in the university, a dear supervisor of mine, Dr. Tim Raeymaekers, welcomed me and introduced me to all my dear friends and colleagues. I still remember Prof. Hendrik De Bie assigned me to my office, which is the largest one in our group and I was the only person in this office for a very long time. I also remember I joined the new semester dinner and a boat trip only several days later! For a new member of the group, I felt the first week was great!

As my life starts here, people around me provided a huge amount of help in both my professional field and daily life. This thesis is a symbol of this four-year journey, it is my great pleasure to use some words to thank the people that always helped me.

First of all, I want to express my deep gratitude to my supervisor Prof. Dr.

Franciscus Sommen, who offered me the great opportunity to study here. He is a brilliant mathematician with a sharing spirit. No matter how hard or easy a question is, he always has a suitable answer and a further direction of exploration.

With this kind help, I am able to tackle all the problems I met in my work. He is also a wonderful person with a lot of interests. I will never forget our conversations about mathematics, history, culture and music. It is a truly privilege to know him and to be his student.

Many thanks to Prof. Dr. Hendrik De Bie who accepted my application in the first place and provided me with all kinds of help in both academic and administrative issues in all these years. He is always accessible to my questions and cares about my work and my daily life, especially during the pandemic period.

I want to thank Dr. Tim Raeymaekers for all his help in these years. He is an earnest supervisor in professional fields and a kind person. He always answers my questions and gives me his suggestions. Under his guidance, I could rapidly fit in the Clifford research group without many difficulties and smoothly pass all these years.

I want to thank Dr. Al´ı Guzm´an Ad´an, who is a very responsible and patient supervisor, especially in the second half of my study here. He shared all his

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mathematical knowledge with me and taught me various techniques in research and writing. His advice really improved my skills in my research field. I also want to thank him and his wife Claudia Matos for inviting me for dinner multiple times. They have great food and atmosphere in their beautiful home.

I’m really grateful to all my supervisors. Without their help, I would not be able to complete this thesis and my study at Ghent University. It is also my pleasure to thank other members in our department who gave me plenty support in these years.

Many thanks to Prof. Dr. Hennie De Schepper for her very kind and friendly help in all these years. As the first person I contacted in the group, I really appreciate her giving me this opportunity to study here. Thanks to Prof. Dr.

Denis Constales for his generous sharing of ideas and great conversations.

I want to thank Dr. Sijia Bao, Teppo Mertens, Sam Claerebout and Ze Yang for their warm company in our office. They are always positive and ready to help to all kinds of questions in both mathematics and daily life. I really appreciate Sijia for her nice and warm help at my beginning days here in Gent, especially for her surprising birthday cake to me. I will miss the days when Teppo and I traveled through Portugal to conferences, he always knows which is the best place to eat and where is the best place to visit. It is my pleasure to be with them in the same office.

Thanks to Srdan, Sigiswald and Wouter who are very enthusiastic and accessible all the time. It is always informative and helpful to talk with them when I encounter problems. Thanks to Astrid and Zo¨e, Michael, Karel, Brecht, Hadewi- jch, Nicolas, Chien, Frederick, Alexis and Asmus for all the nice conversations and great company.

I also want to thank Dr. Amedeo Altavilla who visited the Clifford research group and shared the office with me for a period of time. He is a wonderful person and a great mathematician who also shares a lot about music and cooking. It was my pleasure to visit him in Ancona, Italy. I had a really good time there. Another person I want to thank is Dr. Heikki Orelma, a regular visitor to the group, who can handle multiple languages, including Chinese. He provided many information that I need as a beginner in Gent city. It was also a great opportunity to visit him during a conference in Tampere, Finland. I really enjoyed the delicious smoked salmon fish and the excellent excursions.

Many thanks to Prof. Dr. Irene Sabadini, Prof. Dr. Fabrizio Colombo, Prof.

Dr. Uwe K¨ahler, Prof. Dr. Milton Ferreira and Prof. Dr. Tao Qian. As renowned mathematicians in the field, they spent their valuable time to talk and share their great ideas with me. Their advice and suggestions really helped me a lot in my study.

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v

For all these years, people in the Clifford research group are very nice to me. I will never forget having lunch with my dear friends and colleagues, coffee breaks, fruit time on Tuesday and cake time on Friday. It is my pleasure to be a member in the group.

Meanwhile, I also want to use this opportunity to express my gratitude to many people that support me all the time no matter where I am.

I want to thank my all-time mentor Prof. Dr. Hua Liu, who helped me as much as he can. Without his help and encouragement, I may not be able to go this far. His professional suggestions and earnest advice inspired me in both my study and my way of living. I really appreciate his trust and help. I also want to thank Prof. Dr. Xiaojing Lv for her support and encouragement since my undergraduate study.

Thanks to my best friend Bing Jiang, who always has confidence in me. Even though we didn’t have many chance to get together, he always keeps in touch and cares about my life and my work, no matter how busy he is. The same thanks to my friend Pengfei Gu for his long-distance support. We haven’t met in years, but this doesn’t stop us exchanging ideas and interests. Thanks to an old friend of mine, Zechuan Sun, who always keeps in touch and cares about my status.

Thanks to my friend Yanli Bai, also a great primary school teacher now, for her continuing support and communication in these years.

Special thanks to all my friends here in Gent. Thanks to my great friend Ke Shen for his company in these years. From traveling to shopping, from coding to dining, I really enjoyed the time we have spent together. Also thanks to him and his girl friend Miao Yin for inviting me to dinner multiple times. I really appreciate for all they have done. Thanks to a big brother Liang Ji for his sophisticated advice and suggestions in all these years. We came here at the same time, but he helped me a lot with his experience and wisdom. Thanks to Yaxin, Dengwu and Yongyuan for a wonderful self-driving tour during Christmas in Germany, France and Belgium. It was a great adventure that I have never imagined. Thanks to Chengjun for the time we have spent together. It is my pleasure to know all of them in these years. I really appreciate their company and help.

Last but not least, I want to thank my family for their all-time support no matter how far away I am. My deepest gratitude to my parents Deshui Hu and Zongyu Yin for their unconditional love and support to me. They provided me with their best and always trying to do more. I am really grateful to what they have done and glad to be their child all the time. Thanks to my younger sister Yue Hu for trusting me and always being supportive. Many thanks to my sister Jun Yin and brother Zhi Yin for their great help in all the paperwork I needed.

After more than a year social-distancing and working at home, it is unavoidable

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to mention the pandemic. As a researcher, my life is completely changed during the past year. I miss the days when I can directly talk with my dear friends and colleagues; I miss the days when I can enjoy a cup of drink with my friends; I miss the days when I can visit friends or inviting them for dinner.

All I want now is to wish everyone a safe and healthy life. No matter how hard the time it is now, it will finally pass. When that moment comes, I hope everyone can go back to normal and enjoy their lives again.

Thanks to all of you and wish you all the best!

Ren Hu Gent, February 2021 The doctoral program of the researcher is supported by the China Scholarship Council(CSC) [201708120045].

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4 Bargmann-Radon transform for axially monogenic functions 63 4.1 Monogenic Bargmann module . . . 63 4.2 Bargmann-Radon transform . . . 66 4.3 Bargmann-Radon transform for axially monogenic functions . . . . 69 4.4 Bargmann-Radon transform for CK-extension . . . 74 4.4.1 Explicit formula for CK-extension . . . 74 4.4.2 Bargmann-Radon transform for CK-extension . . . 82 5 Szegö-Radon transform for hypermonogenic functions 87 5.1 Hypermonogenic setting . . . 89 5.1.1 Module of hypermonogenic functions . . . 90 5.1.2 Hypermonogenic plane waves . . . 91 5.1.3 The Szegö-Radon transform in hypermonogenic setting . . 97 5.2 Szegö-Radon transform for hypermonogenic functions . . . 100 5.3 Szegö-Radon transform for the generalized CK-extension . . . 104 5.3.1 Generators ofMH(B(0,1)) . . . 105 5.3.2 Szegö-Radon transform for generalized CK-extension . . . . 109 5.4 The dual of hypermonogenic Szegö-Radon transform . . . 112

Summary 117

Nederlandse samenvatting 119

Bibliography 120

A Appendix 127

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Introduction 1

1.1 Overview

The Radon transform is a well-known tool with many applications in theoretical and applied mathematics. It was originally introduced as an integral of functions on lines on the 2-dimensional plane [1], and later generalized to higher dimensions as an integral over hyperplanes. For a careful study of this transform, some important extensions, and applications we refer the reader to [2, 3, 4, 5, 6, 7]. In this thesis, we will study Radon-type transforms in the setting of Clifford analysis.

In the series of papers [8, 9, 10], F. Sommen initiated this study of Radon- type transforms in the context of Clifford analysis, which is nowadays a well- established research field that generalizes complex analysis to higher dimensions, see e.g. [11, 12, 13, 14]. We make use of Clifford numbers, elements of so-called Clifford algebras, which generalize the classical complex numbers in the sense that a higher number of “complex units” are used. On the other hand, this theory is also a refinement of harmonic analysis, in the sense that one of the key operators in this theory, the so-called Dirac operator, factorizes the Laplace operator, see e.g.

[12, 15, 14]. The main object of study in Clifford analysis are monogenic functions, i.e. null-solutions of the Dirac operator, explicitly defined in this theory as ∂_x = Pm

j=1ej∂xj, which acts on Clifford algebra-valued functions. Here,{e₁, ..., em} is an orthonormal basis ofR^mthat generates the complex Clifford algebraCmby the defining relationse_ie_j+e_je_i =−2δ_ij. Each of these basis elements hence square to

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-1, explaining the generalizing properties of the classical complex numbers stated above.

In the above-mentioned works by Sommen, he studied the extension of the Radon transform to the monogenic setting, i.e. the Clifford-Radon transform, and its relations with the plane wave decompositions of monogenic functions. Many of the results in this thesis will be based on this research.

In this thesis, three concrete settings of Radon type transforms will be studied, namely the Szeg¨o-Radon transforms for classical monogenic functions, its refinement to the hypermonogenic setting and the Bargmann-Radon transform for axially monogenic functions.

1.2 Outline

We start with the basics in Chapter 2. Here, we introduce Clifford algebras (also called geometric algebras), together with definitions, properties and important results that come with these algebras. Standard references are e.g. [11, 12, 16].

We consider functions taking values in these Clifford algebras, in particular those in the kernel of the Dirac operator, the monogenic functions. When studying these kinds of functions, symmetries play a huge role, which is why we consider two special types of monogenic functions with a certain symmetry on their domains, namely the axially monogenic functions and the biaxially monogenic functions.

Moreover, during our reasonings, certain special functions will appear, which will also be defined in this chapter. Next to that, we need some important results in Clifford analysis, namely the Cauchy formula and the Pizzetti formula, see e.g. [17, 18, 19, 20]. Another key result is introduced, the Cauchy-Kovalevskaya extension, which gives a 1-1 connection between analytic functions on a hyperplane and monogenic functions on the full space.

Chapter 3 is devoted to the study of the first Radon type transform of this thesis. In particular, we investigate the action of the Szeg¨o–Radon transform on biaxially monogenic functions. This transform has already been defined in e.g. [21], for general monogenic functions. This Szeg¨o-Radon transform was ab- stractly defined as the orthogonal projection operator of a suitable Hilbert module of monogenic functions on the unit ball, onto a closed submodule of plane waves.

This transformation does not exactly correspond with the Clifford-Radon transform, but it is still a canonical map from m-dimensional monogenic functions to 2-dimensional monogenic functions. Moreover, it offers an alternative to the standard Radon transform for monogenic functions, since the integral of a general monogenic function over a hyperplane does not converge. In this chapter, we go one step further, and consider the Szeg¨o–Radon transform of a special class of

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1.2 Outline 3

monogenic functions, namely those of biaxial symmetry. These are monogenic functions on R^m which are invariant under the subgroup SO(p)×SO(q) of the orthogonal groupSO(m), wherep+q=m. Biaxially symmetric monogenic functions can be obtained by considering solutions of a system of partial differential equations, also known as a Vekua-system, see e.g. [22, 23]. The main target is to compute the Szeg¨o-Radon transform for biaxially monogenic functions.

In Chapter 4, we study the Bargmann-Radon transform. This transform is defined as the projection of the monogenic Bargmann module (of monogenic square integrable functions with Gaussian density) on the closed submodule of monogenic plane waves. Our main goal is to study the action of the Bargmann-Radon transform on a specific type of monogenic functions, namely those on axial domains.

These are null solutions of the Dirac operator with an additional axial symmetry, modeled by a Vekua-type system, e.g. [24, 25, 26, 27, 28].

We determine the explicit action of this transform on axially monogenic functions. Finally, as an example, we determine the action of the Bargmann-Radon transform on functions generated by the CK-extension, a construction technique for monogenic functions.

Finally, in Chapter 5, our goal is to study the refinement of the Szeg¨o-Radon transform to the setting of hypermonogenic functions. These functions were originally introduced in the works of H. Leutwiler and S.-L. Eriksson (see e.g. [29, 30, 31, 32]) as null-solutions of the a modified Dirac operator. The resulting hypermonogenic function theory is related to a hyperbolic upper half-space model.

The reader may find more information and recent research results for example in [32, 33, 34].

In [35], a correspondence between hypermonogenic functions and classical monogenic functions was established. This correspondence allows to define a Hilbert submodule MH(B(0,1)) of hypermonogenic functions inside of the mod- uleML²(B(0,1)). In a similar way, a suitable submoduleMH(s) of hypermonogenic plane wav es arises. The first goal of this chapter is to obtain an explicit expression for the Szeg¨o-Radon transform in the hypermonogenic setting, i.e. the orthogonal projection operator that maps MH(B(0,1)) onto MH(s). We will show that this transform is given by an integral operator which depends on the reproducing kernel of the module MH(s).

We also aim at characterizing the Szeg¨o-Radon image of general hypermonogenic functions. To that end, we explicitly compute the Szeg¨o-Radon projection on a set of generators of MH(B(0,1)). This set of generators is provided by the generalized CK-extension (GCK) of a specific set of generators of a space of analytic functions.

The final purpose of this chapter is to obtain an inversion method for this

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transform. In the classical monogenic setting [21], the Szeg¨o-Radon transform is inverted via a dual transform that involves integration over a Stiefel manifold. In the hypermonogenic setting, this is achieved by means of a dual transform that involves integration over the unit sphere S^q−1. In particular, this dual transformation yields (up to a constant) an inversion method for each of the generators of MH(B(0,1)) mentioned above. The constants appearing in these inversion formulas depend on the homogeneity degrees of the generators, and their explicit computation seems rather involved. For that reason, the problem of finding a global inversion formula for a general hypermonogenic function remains still open.

The results of Chapter 3 and Chapter 4 are published in the papers [36] and [37] respectively. The results of Chapter 5 are submitted for publication [38].

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Basics of Clifford analysis 2

Clifford analysis is nowadays a widely studied subject that has many applications in both pure and applied mathematics. The theory is a powerful generalization of the holomorphic function theory to higher dimension. It deals with the null solutions of the Dirac operator, which are called monogenic functions. The Dirac operator is invariant under the action of the spin group and plays the same role in Clifford analysis as the Cauchy-Riemann operator in complex analysis. Numerous articles and research can be found today with respect to Clifford analysis. We refer the readers to e.g. [11, 12, 16, 39, 40] for a more detailed account on this function theory.

In this chapter, we introduce the classical theory and notions of Clifford analysis which are needed in this thesis. We start with the very basics, i.e. Clifford algebras.

2.1 Preliminaries

Let R^m be the m-dimensional real Euclidean space and let (e₁, e₂, . . . , e_m) be an orthonormal basis for R^m. We denote by Rm the real Clifford algebra generated by these basis elements subjected to the non-commutative multiplication relations

eiej+ejei =−2δ_ij, 1≤i, j≤m, where the symbol δ_ij is the Kronecker delta.

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For any setA={i₁, . . . , i_k} ⊂ {1,2, . . . , m}, ordered by 1≤i₁< . . . < i_k≤m, elements of a canonical basis for Rm are given byeA=ei1ei2. . . eik, wheree∅ = 1 is the identity element. An element a∈Rm can be written as

a=X

A

eAaA, aA∈R.

By grouping the basis elements according to the cardinality|A|=k of the setA, one can also write the element a∈Rm as

a=

m

X

k=0

[a]_k, where

[a]_k= X

|A|=k

e_Aa_A (2.1)

is the so-calledk-vector part ofa. The operator [·]_kprojectsRmonto the subspace of k-vectorsR^(k)m given by

R^(k)m = span{e_i₁e_i₂. . . e_i_k : 1≤i₁ < . . . < i_k≤m}.

Hence, the Clifford algebra Rm admits the multi-vector structure, or multi-vector decomposition

Rm =R⁽⁰⁾m ⊕R⁽¹⁾m ⊕. . .⊕R^(m)m . From the multi-vector structure, one can obtain that

• the space of scalars R⁽⁰⁾m ∼=R has dimension 1;

• the space of (1-)vectors R⁽¹⁾m = span{e₁, . . . , em} has dimensionm and can be identified with R^m;

• the space of 2-vectors, also called bivectors, R⁽²⁾m = span{e_ie_j :i < j} has dimension ^m₂

;

• the space of k-vectorsR^(k)m has dimension ^m_k

;

• the space of m-vectors R^(m)m ∼= Re_M, wheree_M = e1. . . em is the so-called pseudo-scalar, has dimension 1.

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2.1 Preliminaries 7

In particular, we denote byx a vector in Rm which can be expressed as x=

m

X

i=1

eixi.

Then the isomorphism from R⁽¹⁾m to R^m is by virtue of the identification x 7→

(x1, x2, . . . , xm). The product of two vectorsx, y∈Rm can be written as x y=x·y+x∧y,

where

x·y= 1

2(xy+yx) =−

m

X

k=1

xiyi, and

x∧y= 1

2(xy−yx) =X

i<j

(xiyj −xjyi)eiej

are the so-called dot and wedge product respectively. Note that

x·y=−hx, yi, (2.2)

where h·,·idenotes the Euclidean inner product in R^m. It is clear that the inner product of two vectors is scalar and the wedge product of the two is a bivector.

The square of a vector is

x²=−hx, xi=− |x|²

where |x|is the Euclidean norm ofx as a vector of R^m, i.e.

|x|=

m

X

k=1

x²_i

!¹

2

. (2.3)

The multiplication of a vector and a k-vector can be written as xa_k=x·a_k+x∧a_k, a_k∈R^(k)m , where the dot product is defined by the (k−1)-vector part

x·a_k:= 1

2(xa_k−(−1)^ka_kx) (2.4) and the wedge product is defined by the (k+ 1)-vector part

x∧a_k:= 1

2(xa_k+ (−1)^ka_kx). (2.5)

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As any element inRm can be decomposed into multivectors, then the dot product (2.4) and wedge product (2.5) can be extended to a∈Rm as

x·a=

m

X

k=1

x·[a]_k, and x∧a=

m−1

X

k=0

x∧[a]_k. (2.6) We also define the complex Clifford algebra, denoted byCm as the complexi- fication of Rm, i.e.

Cm=C⊗Rm =Rm⊕iRm. Therefore, an element λ∈Cm can be written as

λ=X

A

eAλA, λA∈C.

It is clear that the complex Clifford algebra also has a multi-vector structure.

Moreover, there exist automorphisms on Cm which leave the multi-vector structure invariant. Let λ, µ∈Cm,λA∈C. We have the following automorphisms.

1. The main involution a7→eais defined as

(λµ) =g eλeµ, (λ^_Ae_A) =λ_Aef_A, ee_j =−e_j, j= 1,2, . . . , m.

2. The reversion λ7→λ^∗ is defined as

(λµ)^∗ =µ^∗λ^∗, (λ_Ae_A)^∗=λ_Ae^∗_A, e^∗_j =e_j. 3. The Hermitian conjugationλ→λ^† is defined as

(λµ)^†=µ^†λ^†, (λ_Ae_A)^†=λ^c_Ae^†_A, e^†_j =−e_j, where the λ^c_A is the complex conjugate ofλ_A.

We also define the conjugationa→aonR^m by

ab=ba, aAeA=aAeA, ej =−e_j, a, b∈Rm, aA∈R.

In general, one has that

c^†= (a+ib)^†=a−ib, a, b∈Rm.

The Hermitian conjugation leads to a Hermitian inner product and its associated norm on Cm, given by

(λ, µ) = [λ^†µ]₀, |λ|²= [λ^†λ]₀ = [λ^†λ]₀, λ, µ∈Cm, (2.7) where [.]₀ is the projection operator (2.1) which maps an element a ∈ Cm to its scalar valued part, i.e. [a]0 = a∅. The Hermitian norm in Cm satisfies the property, see [11],

|λµ| ≤2^m/2|λ| |µ|. (2.8)

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2.2 Monogenic functions 9

2.2 Monogenic functions

Monogenic functions play an essential role in Clifford analysis. The notion of monogenicity can be seen as the multi-dimensional counterpart to that of holo- morphy in the complex plane. We start this section by introducing some of the fundamental operators.

Let us denote by P the space ofCm-valued polynomials on R^m, i.e.

P =C[x1, . . . , xm]⊗Rm.

One of the most important operators in Clifford analysis is the so-called Dirac operator

∂x =

m

X

j=1

ej∂xj,

which is a first order differential operator. Let Ebe the so-called Euler operator defined by

E=hx, ∂_xi=

m

X

j=1

xj∂xj, (2.9)

wherehx, ∂_xidenotes the Euclidean inner producthx, yiby formally substituting all variables y_j iny by ∂_x_j. The Gamma operator is defined by

Γ =−x∧∂_x =−X

i<j

e_ij(x_i∂_x_j−x_j∂_x_i), (2.10)

where Lij =xi∂xj−xj∂xi,i, j= 1,2, . . . , m, are the well-known angular momentum operators. It can be easily seen that

Γ =−x∂_x−E. (2.11)

Let us denote by r the norm of the vector x in (2.3), i.e. r = |x|. Then the following commutation rules hold for these operators.

{x, ∂_x}=−2E−m, [E, x] =x, [E, ∂_x] =−∂_x, [E,Γ] = 0,

[Γ, x^2k] = 0, {Γ, x^2k+1}= (m−1)x^2k+1,

(2.12)

where {a, b}:=ab+baand [a, b] :=ab−ba.

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The Dirac operator can be seen not only as the higher dimensional counterpart of the Cauchy-Riemann operator in complex analysis but also as the “square root”

of the Laplace operator which is given by

∆x =−∂_x² =

m

X

j=1

∂_x²_j. (2.13)

The following commutation rules hold with respect to the Laplace operator, [∆_x, r²] = 2(2E+m), [∆_x,E+m

2] = 2∆_x. (2.14) Letf be a function on an open subset Ω⊂R^m of the form

f =X

A

f_Ae_A,

where the functions f_A are R-valued or C-valued. Then we can introduce monogenic functions and harmonic functions.

Definition 2.1 (Monogenic functions). A function f : Ω→Rm is called left- or right- monogenic if it is a null-solution of the Dirac operator, i.e.

∂_xf = 0 or f ∂_x= 0. (2.15)

Remark 2.2. Left-monogenic functions and right-monogenic functions are con- nected through the Hermitian conjugation, i.e.

(∂xf)^†=−f^†∂x. (2.16)

Therefore, it is sufficient for us to only consider left-monogenic functions in our work. From now on left-monogenic functions will be called monogenic functions unless it is stated otherwise.

We denote byM(Ω) the space of left-monogenic functions on an open subset Ω⊂R^m.

Definition 2.3(Harmonic functions). A functionh: Ω→Rm is called harmonic if it is a null-solution of the Laplace operator, i.e.

∆_xh= 0.

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We denote byP_k the space of homogeneous polynomials of degreek,k∈N: P_k :={R∈ P:R(λx) =λ^kR(x),∀λ∈C}. (2.17) Similarly, we can define homogeneous functions of any degree of homogeneity s∈Rby requiring f(λx) =λ^sf(x).

The space of polynomials can be decomposed as P =

∞

M

k=0

P_k.

One can easily obtain that the Euler operator measures the degree of homogeneity of polynomials R_k(x)∈ P_k, i.e.

ERk(x) =kRk(x). (2.18)

In other words, homogeneous polynomials of degree k are the eigenfunctions of the Euler operator with eigenvalue k. ThusP_k can also be written as

P_k={R∈ P :ER=kR}.

Note that when we want to stress the domain of the polynomials, we use the no- tationP_k(R^m) instead ofP_k. LetM⁺_k be the space of inner spherical monogenics of degree k∈N:

M⁺_k :={R∈ P_k:∂_xR= 0}. (2.19) The space of spherical harmonics of degree kis defined by

H_k:={R ∈ P_k : ∆xR= 0}.

Due to the factorization (2.13) of the Laplace operator, one may say Clifford analysis constitutes a refinement of harmonic analysis. Furthermore, we have M⁺_k ⊂ H_k ⊂ P_k. Moreover, let M_k ∈ M⁺_k, then (2.18) and (2.11) directly leads to the properties

ΓM_k=−kM_k. (2.20)

The vector variablex∈R^m can be written in spherical coordinates as

x=rω, (2.21)

where r=|x| ∈[0,+∞[ and ω∈S^m−1={x∈R^m:|x|= 1}. The Dirac operator in spherical coordinates is given by

∂_x =ω(∂_r+1 rΓ).

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The Euler operator can be written as E=

m

X

j=1

x_j∂_x_j =r∂_r. The following commutation rules hold

E, 1

r²

=−2

r², [Γ, r²] = 0.

Meanwhile, the Laplace operator in spherical coordinates is given by

∆_x =∂_r²+ m−1

r ∂_r+ 1

r²∆_ω, (2.22)

where ∆_ω = Γ(m−2−Γ) is the so-called Laplace-Beltrami operator on S^m−1. Due to the fact that ∆x,∂r and r² are all scalar operators, the Laplace-Beltrami operator has to be scalar. Therefore,

∆ω= [Γ(m−2−Γ)]₀=− Γ²

0 =X

i<j

L²_ij,

where Lij are the angular momentum operators defined in (2.10). LetH_k ∈ H_k, then from the identity

∆_xH_k=

∂_r²+ m−1

r ∂_r+ 1 r²∆_ω

H_k= k²+k(m−2)

H_k+ ∆_ωH_k = 0, we obtain that elements inH_kare eigenfunctions of the Laplace-Beltrami operator with the eigenvalue of −k(k+m−2).

2.2.1 Fischer decomposition

One fundamental result from spherical monogenics is the so-called Fischer decomposition, see e.g. [12] p.204. The Fischer decomposition is applied in this thesis to investigate the structure of monogenic and hypermonogenic functions. We start this section with the so-called Fischer inner product.

Definition 2.4 (Fischer inner product). Let R(x), S(x) ∈ P. Then the Fischer inner product is given by

hR, Si_∂:=

h

R(∂x)^†S(x) i

0

x=0,

where R(∂x) is the so-called the Fischer dual ofR(x)in which all variablesxj are replaced by ∂_x_j.

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In order to explain the Fischer decomposition, we recall first the following result from linear algebra. Let U and V be finite dimensional vector spaces with inner productsh·,·i_U andh·,·i_V respectively. LetA:U →V be a linear map and let A^∗ be the adjoint ofA, i.e.

hAu, vi_V =hu, A^∗vi_U, ∀u∈U,∀v∈V.

It can be proven that (ImA)^⊥= ker(A^∗). Thus we obtain V = ImA⊕ker(A^∗).

Lets, k∈N,s < k, andP ∈ P_s. We define the multiplication operator associated with P by

P·:P_k−s→ P_k:R 7→P R.

It is easily seen that P(∂x)^† is the adjoint operator of P· with respect to the Fischer inner product. Consequently, we have

P_k = ker(P(∂x)^†)⊕P(x)(P_k−s),

which is the so-called Fischer decomposition. In particular, taking P = x and P =|x|² yields

P_k=M⁺_k ⊕xP_k−1 and P_k=H_k⊕r²P_k−2

respectively. These results lead to the following two theorems which play an important role in this thesis, see e.g. [12, 41].

Theorem 2.5 (Monogenic Fischer decomposition). Let k∈N, then the space P_k decomposes as follows

P_k=

k

M

j=0

x^jM⁺_k−j,

i.e. for each R_k∈ P_k, there exist uniqueMk−j ∈ M⁺_k−j such that R_k=

k

X

j=0

x^jMk−j(x).

Theorem 2.6 (Harmonic Fischer decomposition). Let k∈N. Then P_k=

b^k

2c

M

j=0

|x|^2jH_k−2j, where b·cis the floor function.

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In particular, the space of harmonic functions H_k can be decomposed imme- diately following from the above theorems, which leads to the next proposition.

Proposition 2.7. Let k∈N, the decomposition holds

H_k=M⁺_k ⊕xM⁺_k−1, (2.23)

where for every H_k ∈ H_k, there exist unique monogenic elements M_k ∈ M⁺_k and Mk−1 ∈ M⁺_k−1 such that

M_k= k+m−2−Γ 2k+m−2 H_k, xMk−1 = k+ Γ

2k+m−2H_k.

One of the most important tools related to spherical harmonics is the so-called Funk-Hecke Theorem, see e.g. [27, 42, 43]. It was first introduced by Funk and Hecke in [44, 45, 46, 47] in a 3-dimensional settings and then by A. Erd´elyi in [42]

in n-dimensional space.The Funk-Hecke Theorem is an essential tool to compute spherical integrals related to Radon type transforms.

Theorem 2.8 (Funk-Hecke theorem). Let ξ, η ∈ S^m−1. Let ψ be a real-valued function whose domain contains [−1,1] and let S_k(ξ) be a spherical harmonic function of degree k. Then we have

Z

S^m−1

ψ(hξ, ηi)S_k(η)dS(η)

= k!Am−1

(m−2)k

S_k(ξ) Z 1

−1

ψ(t)C

m 2−1

k (t)(1−t²)^(m−3)/2dt.

(2.24)

where dS(η) is the scalar element of surface area on S^m−1, C_m^λ(t) is the Gegen- bauer polynomial, Am = ^2π_Γ(^m/2m

2) is the area of the unit sphere S^m−1 and (a)k = a(a+ 1)· · ·(a+k−1) is the Pochhammer symbol.

2.2.2 Inner and outer spherical monogenics

We denote by ˚B(0, R) (B(0, R)), R > 0, the open (closed) ball centered at the origin with radius R. Letx∈R^m\{0}, the inverse ofx is given by

x7→ − x

|x|²,

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which defines a bijection between ˚B(0, R) andR^m\B(0,1/R). The action of the Dirac operator under this change of coordinates leads to the so-called Clifford inversion I, which is an operator preserving monogenicity. It is given by:

I[f](x) = x

|x|^mf x

|x|²

.

This inversion operator has the following properties (see [12] p.157)

1. Iff ∈ M( ˚B(0, R)), thenI[f]∈ M(R^m\B(0,1/R)) and limx→∞I[f](x) = 0;

2. I²[f] =−f;

3. ∂x[If] =− |x|²I[∂xf].

The Clifford inversion is an essential tool for investigating the local behavior of monogenic functions in SO(m)-invariant domains. In this setting, the spaces of inner and outer spherical monogenics play an essential role.

Definition 2.9. We have the following definitions.

1. We recall that M⁺_k is the space of homogeneous monogenic polynomials of degree k in R^m, see (2.19). An arbitrary element of it, usually denoted by Pk, is called inner spherical monogenic of order k.

2. LetM⁻_k be the space of homogeneous monogenic functions of degree−(m+ k−1) in R^m\{0}. An arbitrary element of it, usually denoted by Q_k, is called outer spherical monogenic of order k.

3. Let M^±_k(S^m−1) be the space of the restrictions of functions in M^±_k to the unit sphere S^m−1. Arbitrary elements of M⁺_k and M⁻_k are usually denoted by Pk(ω) and Qk(ω) respectively.

4. The spaceM_k(S^m−1) of spherical monogenics of order k is defined as M_k(S^m−1) =M⁺_k(S^m−1) +M⁻_k(S^m−1).

5. Let H_k(S^m−1) be the space of restrictions of H_k to the unit sphere S^m−1. From (2.23) it follows that

H_k(S^m−1) =M⁺_k(S^m−1) +ωM⁺_k−1(S^m−1).

Note that when we want to stress the domain of the polynomial, we use the notation M⁺_k (R^m) instead of M⁺_k.

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Remark 2.10. Obviously,M^±_k andM^±_k(S^m−1) are rightCm-modules.

The inversion I establishes an isomorphism I : M⁺_k → M⁻_k. Indeed, let P_k∈ M⁺_k, then

I[Pk](x) = x

|x|^mPk

x

|x|²

has degree of homogeneity equal to (−k−m+ 1), which shows thatI[P_k]∈ M⁻_k. On the other hand, if Qk∈ M⁻_k, then

I[Q_k](x) = x

|x|^mQ_k x

|x|²

has degree of homogeneity equal to k, i.e. I[Q_k] ∈ M⁺_k. An element Q_k(ω) ∈ M⁻_k(S^m−1) is the restriction to the sphere of Qk∈ M⁻_k, which can be written as

Q_k(x) =I[P_k](x) = x

|x|^mP_k x

|x|²

. The above formula implies that

Q_k(ω) =ωP_k(ω),

which is the restriction of the polynomial xP_k(x)∈ H_k+1. Then we obtain M⁻_k(S^m−1) =ωM⁺_k(S^m−1)⊂ H_k+1(S^m−1).

2.2.3 Decomposition of L₂(S^m−1)

Let us denote byL₂(S^m−1) the space of all Clifford-valued square integrable functions on the unit sphere S^m−1. One can define the inner product on this space by

hf, gi_S= 1 A_m

Z

S^m−1

g^†(ω)f(ω)dS(ω), (2.25)

which is a Cm-valued integral. Moreover, we define hf, gi₀ = [hf, gi_S]₀= 1

A_m Z

S^m−1

h

g^†(ω)f(ω)i

0dS(ω). (2.26) Theorem 2.11. Let Pk ∈ M⁺_k, P` ∈ M⁺_` , Qk ∈ M⁻_k and Q` ∈ M⁻_` , then we have the following properties:

• hP_k, P_`i_S=hQ_k, Q_`i_S= 0, if k6=`.

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• hP_k, Q_`i_S= 0, for allk, `.

• H_k(S^m−1) =M⁺_k(S^m−1)⊕ M⁻_k−1(S^m−1).

From Proposition 2.7, we have the following theorem.

Theorem 2.12. EveryHk(ω)∈ H_k(S^m−1) admits a unique orthogonal decomposition

H_k =PkH_k(ω) +Qk−1H_k(ω), (2.27) where Pk and Qk−1 are projection operators onto M⁺_k(S^m−1) and M⁻_k−1(S^m−1) respectively, acting on H_k(ω) as

Pk(Hk)(ω) = k+m−2−Γ

2k+m−2 Hk(ω), Qk−1(Hk)(ω) = k+ Γ

2k+m−2Hk(ω).

Now let f ∈ L₂(S^m−1). It is a known result that f admits a decomposition into spherical harmonics

f(ω) =

∞

X

k=0

Sk(f)(ω),

whereSk is the orthogonal projection operatorSk:L₂(S^m−1)→ H_k(S^m−1) which is given by

Sk(f)(ω) = N(m;k) Am

Z

S^m−1

Pk,m(hω, ηi)f(ω)dω, whereN(m;k) = ^k+m−1_k

+ ^k+m−2_k−1

is the dimension (over the Clifford algebra) of the spaceH_k(S^m−1) andPk,mis the Legendre polynomial (2.67) of degree kin m dimensions.

Using the decomposition (2.27), Sk(f)∈ H_k(S^m−1) can be written as Sk(f) =Pkf+Qk−1f,

where the actions of Pk and Qk are extended, by a slight abuse of notation, to f ∈ L₂(S^m−1) by means of

Pkf =PkSk(f) = k+m−2−Γ 2k+m−2 Sk(f), Qkf =QkSk+1(f) = k+ 1 + Γ

2k+m Sk+1(f).

Obviously, we have

Pk:L₂(S^m−1)→ M⁺_k(S^m−1), and Qk:L₂(S^m−1)→ M⁻_k(S^m−1).

Let us denote Π_k=Pk+Qk−1. We thus have the following theorem.

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Theorem 2.13 (Orthogonal decomposition of L₂(S^m−1)). Let f ∈ L₂(S^m−1), then f admits a decomposition into spherical monogenics

f =

∞

X

k=0

Pkf +Qk−1f =

∞

X

k=0

Π_kf. (2.28)

Moreover,

|f|² =

∞

X

k=0

|Π_kf|² =

∞

X

k=0

|Pkf|+|Qk−1f|²,

where |·| is the L₂-norm associated to the inner product (2.26). In general, if g∈ L₂(S^m−1), one has

hf, gi_S=

∞

X

k=0

hPkf ,Pkgi_S+hQkf ,Qkgi_S.

2.2.4 Axially monogenic functions

As a subclass of the monogenic functions, axially monogenic functions were introduced in [48] and further studied in e.g. [49, 50, 22]. They are monogenic functions in R^m+1 defined in a symmetric domain and satisfying special plane elliptic systems.

LetR^m+1 denote the (m+ 1)-dimensional Euclidean space where~x∈R^m+1 is written as

~

x=x0+x, x0∈R, x∈R⁽¹⁾_m .

An open domain Ω⊂R^m+1 is called axially symmetric if it is invariant under the group of rotations SO(m) leaving the x₀-axis invariant, i.e.

~

x=x₀+x∈Ω→x₀+M x∈Ω, ∀M ∈SO(m).

Therefore, we may introduce cylindrical coordinates (x0, ρ, η) in Ω by means of the formulas

~

x=x₀+ρη, ρ=|x|, η= x

|x|.

Furthermore there exists a set ˜Ω in the half plane {(x₀, ρ) :ρ≥0} such that Ω ={x₀+ρη: (x₀, ρ)∈Ω, η˜ ∈S^m−1}.

Letf : Ω→Cmbe a left-monogenic function with respect to the Cauchy-Riemann operator∂x0 +∂x in Ω, i.e.

(∂_x₀ +∂_x)f = 0.

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Then in view of the orthogonal decomposition (2.28) of L₂(S^m−1) , for each (x0, ρ)∈Ω fixed, we expand˜ f into a series of spherical monogenics onS^m−1

f(x₀, ρ, η) =

∞

X

k=0

Pkf_(x₀_,ρ)(η) +Qkf_(x₀_,ρ)(η) =

∞

X

k=0

Π_kf(~x), (2.29) where

Π_kf(~x) =Pkf_(x₀_,ρ)(η) +Qkf_(x₀_,ρ)(η).

In [48] it was proven that the above series constitutes a generalized Laurent expansion for monogenic functions with respect to ∂x0 +∂x in Ω.

Theorem 2.14 (Laurent expansion in axially symmetric domain). The series (2.29) converges uniformly on compact sets of Ω and the functions Π_kf(~x) are left monogenic on Ω, i.e.

(∂_x₀ +∂_x)Π_kf(~x) = 0.

For each (x₀, ρ)∈Ω fixed, we write˜

Π_kf(~x) =A_k,η(x0, ρ) +ηB_k,η(x0, ρ), where we have introduced the notations

A_k,η(x0, ρ) =Pkf_(x₀_,ρ)(η), and

Bk,η(x0, ρ) =−ηQkf_(x₀_,ρ)(η).

Observe thatAk,η(x0, ρ) andBk,η(x0, ρ) are inner spherical monogenics of degreek with respect toη. This particular form of the summands of the Laurent expansion leads to the following definition.

Definition 2.15 (Axially monogenic functions). Let f be a left monogenic function in Ω, then f is called axially monogenic if it is of the form

f(x₀, x) = A(x₀, ρ) +ηB(x₀, ρ)

M_k(η), (2.30)

where ∂_x₀+∂_x

f(x₀, x) = 0.

The Cauchy-Riemann operator can be written in cylindrical coordinates as

∂_x₀ +∂_x=∂_x₀ +η∂_ρ+1 ρηΓ.

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Thus, using the monogenicity off(x₀, x), we obtain that functionsA andB from (2.30) satisfy the Vekua-type system











∂x0A−∂ρB = ^k+m−1_ρ B,

∂_x₀B+∂_ρA = ^k_ρA.

(2.31)

Example 2.16. A good example of axially monogenic functions is given by functions of exponential type, i.e. functions of the form

E_k(x0, x)M_k(η) = A(x0, ρ) +ηB(x0, ρ)

M_k(η).

with A(x0, ρ) = e^x⁰a(ρ) and B(x0, ρ) = e^x⁰b(ρ). In this case, the system (2.31) becomes











a(ρ)−b⁰(ρ) = ^k+m−1_ρ b(ρ), b(ρ) +a⁰(ρ) = ^k_ρa(ρ).

From the second equation we obtain

b⁰(ρ) =−a⁰⁰(ρ) + ka⁰(ρ)

ρ −ka(ρ) ρ² .

Substituting b⁰(ρ) and b(ρ) into the first equation yields the ordinary differential equation

a⁰⁰(ρ) +m−1

ρ a⁰(ρ) +

−k(k+m−2)

ρ² + 1

a(ρ) = 0.

The solutions of this equation are of the form, see e.g. [27, 49]

a(ρ) =Cρ¹⁻^m²J_k+^m

2−1(ρ), where C∈R is a constant and

J_α(ρ) =

∞

X

j=0

(−1)^j j!Γ (j+α+ 1)

ρ 2

2j+α

is the Bessel function of the first kind, see e.g. [27]. Thus we have b(ρ) =Cρ¹⁻^m²J_k+^m

2(ρ).

We now put

E_k(x₀, x)M_k(η) =Ce^x⁰ρ¹⁻^m²(J_k+^m

2−1(ρ) +ηJ_k+^m

2(ρ))M_k(η), where C= 2^k+^m²⁻¹Γ k+^m₂

. Note that

E_k(x₀,0) =e^x⁰.

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2.2.5 Biaxially monogenic functions

Let R^m =R^p⊕R^q, with m =p+q. Then any elementx∈R^p+q can be written as

x=x₁+x₂ (2.32)

for all x₁ ∈R^p and x₂ ∈R^q. Let Ω be an open domain inR^p+q, then we define Ω to be a biaxial domain if it is invariant under SO(p)×SO(q), i.e.

x=x₁+x₂ ∈Ω→M₁x₁+M₂x₂ ∈Ω,

where M₁ ∈SO(p) andM₂ ∈SO(q). The Dirac operator inR^p+q can be written as

∂x=∂x₁ +∂x₂, (2.33)

where

∂x₁ =

p

X

i=1

ei∂xi, ∂x₂ =

m

X

j=p+1

ej∂xj. Note that x₁x₂ = −x₂x₁ and ∂_x₁∂_x₂ = −∂_x

2∂_x₁. Accordingly, we define the Gamma operators

Γ_i =−x_i∧∂_x_i, i= 1,2. (2.34) Using spherical coordinates, one can write

x=r₁ω₁+r₂ω₂, ω₁∈S^p−1, ω₂ ∈S^q−1, where r₁ =|x₁|,r₂ =|x₂|. Then we have

∂_x_j =ω_j

∂_r_j + 1 r_jΓ_j

, j= 1,2, (2.35)

which also shows that Γ1 and Γ2 commute.

Moreover, we consider the so-called bispherical monogenics, denoted byf(ω₁, ω₂), which are simultaneous eigenfunctions of Γ₁ and Γ₂. In view of [51, 52], the eigenvalues of (Γ1,Γ2) can only be pairs of the form (−k,−`), (k+p−1,−`), (−k, `+q−1) and (k+p−1, `+q−1), where k, ` ∈ N. The corresponding eigenspaces are denoted as follows.

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eigenspace eigenvalues M^0,0_k,` (−k,−`) M^1,0_k,` (k+p−1,−`) M^0,1_k,` (−k, `+q−1) M^1,1_k,` (k+p−1, `+q−1) Besides, since for any f(ω₁, ω₂), we have (see (2.12))











Γ1ω₁f(ω₁, ω₂) =ω₁(p−1−Γ1)f(ω₁, ω₂), Γ₂ω₂f(ω₁, ω₂) =ω₂(q−1−Γ₂)f(ω₁, ω₂),

(2.36)

it is easily seen that the elements of these spaces are of the form A(ω₁, ω₂), ω₁B(ω₁, ω₂), ω₂C(ω₁, ω₂) and ω₁ω₂D(ω₁, ω₂), where A, B, C and D belong to M^0,0_k,`.

Proposition 2.17 (see [22]). A function f(ω₁, ω₂) belongs to M^0,0_k,` if and only if it is the restriction to the bisphere S^p−1×S^q−1 of a homogeneous polynomial f(x₁, x₂) of degree k in x₁ and of degree `in x₂, satisfying the relation

∂x₁f(x₁, x₂) =∂x₂f(x₁, x₂) = 0.

Definition 2.18. Let us define M_k,` to be the total space of bispherical monogenics of degree (k, `) which is given by (see [22])

M_k,`=M^0,0_k,`⊕ M^1,0_k,`⊕ M^0,1_k,`⊕ M^1,1_k,`,

and H_k,` to be the space of functions f(ω₁, ω₂) which are spherical harmonic of degree k in ω₁ and `in ω₂ which is given by

H_k,`=M^0,0_k,`⊕ M^1,0_k−1,`⊕ M^0,1_k,`−1⊕ M^1,1_k−1,`−1.

Similar to the axially monogenic case in the previous section, we are interested in obtaining a Laurent expansion for monogenic functions in biaxially symmetric domains. To that end, we need the following direct consequence of Theorem 2.13.

Radon type transforms in Clifford analysis

Radon type transformaties in cliffordanalyse