Hyperholomorphic structures and corresponding explicit orthogonal function systems in 3D and 4D By the Faculty of Mathematik und Informatik of the Technische Universität Bergakademie Freiberg approved Thesis to attain the academic degree of Doctor rerum naturalium (Dr. rer. nat.) submitted by M.Sc. Thu Hoai, Le born on the 23. December 1981 in Haiphong-Vietnam Assessor: Prof. Dr. rer. nat. habil. Wolfgang Sprößig, Germany Prof. Dr. rer. nat. habil. Klaus Gürlebeck, Germany Dr. rer. nat. João Pedro Leitão da Cruz Morais, Portugal Date of the award:Freiberg, 20th June 2014 Versicherung Hiermit versichere ich, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe; die aus fremden Quellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht. Bei der Auswahl und Auswertung des Materials sowie bei der Herstellung des Manuskripts habe ich Unterstützungsleistungen von folgenden Personen erhalten: ◦ Professor Dr. rer. nat. habil. Wolfgang Sprößig ◦ Dr. rer. nat. João Pedro Leitão da Cruz Morais Weitere Personen waren an der Abfassung der vorliegenden Arbeit nicht beteiligt. Die Hilfe eines Promotionsberaters habe ich nicht in Anspruch genommen. Weitere Personen haben von mir keine geldwerten Leistungen für Arbeiten erhalten, die nicht als solche kenntlich gemacht worden sind. Die Arbeit wurde bisher weder im Inland noch im Ausland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörde vorgelegt. 20th June 2014 M.Sc. Thu Hoai, Le Declaration I hereby declare that I completed this work without any improper help from a third party and without using any aids other than those cited. All ideas derived directly or indirectly from other sources are identified as such. In the selection and use of materials and in the writing of the manuscript I received support from the following persons: ◦ Professor Dr. rer. nat. habil. Wolfgang Sprößig ◦ Dr. rer. nat. João Pedro Leitão da Cruz Morais Persons other than those above did not contribute to the writing of this thesis. I did not seek the help of a professional doctorate-consultant. Only those persons identified as having done so received any financial payment from me for any work done for me. This thesis has not previously been published in the same or a similar form in Germany or abroad. 20th June 2014 M.Sc. Thu Hoai, Le 5 Zusammenfassung Die Reichhaltigkeit und breite Anwendbarkeit der Theorie der holomorphen Funktionen in der komplexen Ebene ist stark motivierend eine ähnliche Theorie für höhere Dimensionen zu entwick- eln. Viele Forscher waren und sind in diese Aufgaben involviert, insbesondere in der Entwicklung der Quaternionenanalysis. In den letzten Jahren wurde die Quaternionenanalysis bereits erfol- greich auf eine Vielzahl von Problemen der mathematischen Physik angewandt. Das Ziel der Dissertation besteht darin, holomorphe Strukturen in höheren Dimensionen zu studieren. Zunächst wird ein neues Holomorphiekonzept vorgelegt, was auf der Theorie rechts- invertierbarer Operatoren basiert und nicht auf Verallgemeinerungen des Cauchy-Riemann-Systems wie üblich. Dieser Begriff umfasst die meisten der gut bekannten holomorphen Strukturen in höheren Dimensionen. Unter anderem sind die üblichen Modelle für reelle und komplexe quaternionenwertige Funktionen sowie Clifford-algebra-wertige Funktionen enthalten. Außerdem werden holomorphe Funktionen mittels einer geeignete Formel vom Taylor-Typ durch spezielle Funktionen lokal approximiert. Um globale Approximationen für holomorphe Funktionen zu erhalten, werden im zweiten Teil der Arbeit verschiedene Systeme holomorpher Basisfunktionen in drei und vier Dimensionen mittels geeigneter Fourier-Entwicklungen explizit konstruiert. Das Konzept der Holomorphie ist verbunden mit der Lösung verallgemeinerter Cauchy-Riemann Systeme, deren Funktionswerte reellen Quaternionen bzw. reduzierte Quaternionen sind. In expliziter Form werden orthogonale holomorphe Funktionensysteme konstruiert, die Lösungen des Riesz-Systems bzw. des Moisil- 3 Teodorescu Systems über zylindrischen Gebieten im R , sowie Lösungen des Riesz-Systems in 4 Kugeln des R sind. Um konkrete Anwendungen auf Randwertprobleme realisieren zu können wird eine orthogonale Zerlegung eines Rechts-Quasi-Hilbert-Moduls komplex-quaternionischer Funktionen unter gegebenen Bedingungen studiert. Die Ergebnisse werden auf die Behandlung von Maxwell-Gleichungen mit zeitvariabler elektrischer Dielektrizitätskonstante und magnetis- cher Permeabilität angewandt. 7 Abstract The richness and widely applicability of the theory of holomorphic functions in complex analysis requires to perform a similar theory in higher dimensions. It has been developed by many researchers so far, especially in quaternionic analysis. Over the last years, it has been successfully applied to a vast array of problems in mathematical physics. The aim of this thesis is to study the structure of holomorphy in higher dimensions. First, a new concept of holomorphy is introduced based on the theory of right invertible operators, and not by means of an analogue of the Cauchy-Riemann operator as usual. This notion covers most of the well-known holomorphic structures in higher dimensions including real, complex, quaternionic, Clifford analysis, among others. In addition, from our operators a local approximation of a holomorphic function is attained by the Taylor type formula. In order to obtain the global approximation for holomorphic functions, the second part of the thesis deals with the construction of different systems of basis holomorphic functions in three and four dimensions by means of Fourier analysis. The concept of holomorphy is related to the null-solutions of generalized Cauchy-Riemann systems, which take either values in the reduced quaternions or real quaternions. We obtain several explicit orthogonal holomorphic function sys- 3 tems: solutions to the Riesz and Moisil-Teodorescu systems over cylindrical domains in R , and 4 solutions to the Riesz system over spherical domains in R . Having in mind concrete applications to boundary value problems, we investigate an orthogonal decomposition of complex-quaternionic functions over a right quasi-Hilbert module under given conditions. It is then applied to the treat- ment of Maxwell’s equations with electric permittivity and magnetic permeability depending on the time variable. 9 Acknowledgements This work has been carried out during the years 2009-2014 at the Institute of Applied Mathe- matics, the Faculty of Mathematics and Computer Science, the Freiberg University of Mining. The work presented in this thesis would not have been possible without my close association with many people. I take this opportunity to extend my sincere gratitude and appreciation to all those who made this Ph.D thesis possible. First and foremost, I owe my deepest gratitude to my supervisor Prof. Dr. rer. nat. habil. Wolfgang Sprößig, who led me into this interesting topic and always worked with me closely. Without his continuous optimism concerning this work, enthusiasm, encouragement and support this study would hardly have been completed. I also express my warmest gratitude to my other supervisor Dr. rer. nat. João Pedro Leitão da Cruz Morais. His guidance, great co-work and very detailed correction have been essential during this work. I am very thankful for their constant help, enthusiastic guidance, patience, and encouragement. It has been an honor for me to be the last Ph.D. student of Prof. W. Sprößig as well as the first Ph.D student of Dr. J. Morais. I am deeply grateful to Prof. Klaus Gürlebeck, Prof. Michael Shapiro, Prof. Elena Luna Elizarraras, Prof. Sören Kraußhar, Prof. Richard Delanghe and Prof. Michael Eiermann for their scientific advice and knowledge and many insightful discussions and suggestions about my topic. I am indebted to the staff and my colleagues in the Institute of Applied Analysis, the Faculty of Mathematics and Computer Science, especially Prof. Elias Wegert and Frau Karin Uhlemann, who always support me throughout my time. I would like to express my thanks to them all very sincerely. My thanks go to my colleagues in the School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Vietnam for their administrative help and good wishes. Especially, I am very grateful to Prof. Le Hung Son for his encourage and valuable suggestion during this study. A special mention of thanks to my friends in Germany and in Vietnam, who have always en- couraged me. I express my heart-felt gratitude to my family, my mom, my dad and my sister for their uncon- ditional love and care during my life. I truly thank my husband for sticking by my side, even when I was irritable and depressed. My love to my little son for giving me happiness during the last two and a half years of my studies. In conclusion, I recognize that this research would not have been possible without the financial support during my work. I acknowledge financial support from the Deutscher Akademischer Austausch Dienst (DAAD) via the PhD/grant A/08/74960. Finally, I would like to thank Frau Elke Burbach and other staff members of DAAD for their valuable advices and supports on living in Germany. 11 Contents Introduction 13 1 Preliminaries of Quaternionic Analysis 21 1.1 Real and Complex Quaternionic Analysis . 22 1.1.1 Real Quaternions . 22 1.1.2 Complex Quaternions . 24 1.1.3 Reduced Quaternions . 29 1.2 Multi-dimensional Generalizations of the Cauchy-Riemann Operator . 30 1.2.1 A Generalized