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Hypercomplex Analysis: New Perspectives and Applications Trends in Mathematics Swanhild Bernstein Uwe Kähler Irene Sabadini Frank Sommen Editors Hypercomplex Analysis: New Perspectives and Applications Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference. More information about this series at http://www.springer.com/series/4961 Hypercomplex Analysis: New Perspectives and Applications Swanhild Bernstein Uwe Kähler Irene Sabadini Frank Sommen Editors Editors Swanhild Bernstein Uwe Kähler Institute of Applied Analysis Departamento de Matemática TU Bergakademie Freiberg Universidade de Aveiro Freiberg, Germany Aveiro, Portugal Irene Sabadini Frank Sommen Dipartimento di Matematica Dept. Mathematical Analysis Politecnico di Milano University of Gent Milano, Italy Gent, Belgium ISSN 2297-0215 ISSN 2297-024X (electronic) ISBN 978-3-319-08770-2 ISBN 978-3-319-08771-9 (eBook) DOI 10.1007/978-3-319-08771-9 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014952606 Mathematics Subject Classification (2010): 30G35, 30G25, 22E46, 32A50, 68U10 © Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad- casting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com) Contents Preface .................................................................. vii R. Abreu Blaya, J. Bory Reyes, A. Guzm´an Ad´an and U. K¨ahler Symmetries and Associated Pairs in Quaternionic Analysis . 1 D. Alpay, F. Colombo and I. Sabadini Generalized Quaternionic Schur Functions in the Ball and Half-spaceandKrein–LangerFactorization .......................... 19 D. Alpay, F. Colombo, I. Sabadini and G. Salomon TheFockSpaceintheSliceHyperholomorphicSetting .............. 43 E. Ariza and A. Di Teodoro Multi Mq-monogenicFunctioninDifferentDimension ............... 61 S. Bernstein TheFractionalMonogenicSignal .................................... 75 L.J. Carmona L., L.F. Res´endis O. and L.M. Tovar S. WeightedBergmanSpaces .......................................... 89 D. Eelbode and N. Verhulst On Appell Sets and Verma Modules for sl (2) ....................... 111 S.-L. Eriksson, H. Orelma and N. Vieira Integral Formulas for k-hypermonogenic Functions in R3 ............ 119 R. Ghiloni, V. Moretti and A. Perotti Spectral Properties of Compact Normal Quaternionic Operators . 133 Yu. Grigor’ev Three-dimensional Quaternionic Analogue of the Kolosov–MuskhelishviliFormulae ................................... 145 K. G¨urlebeck and D. Legatiuk On the Continuous Coupling of Finite Elements withHolomorphicBasisFunctions .................................. 167 vi Contents K. G¨urlebeck and H. Manh Nguyen On ψ-hyperholomorphic Functions and a DecompositionofHarmonics ...................................... 181 U. K¨ahler and N. Vieira FractionalCliffordAnalysis ......................................... 191 Y. Krasnov Spectral Properties of Differential Equations in Clifford Algebras . 203 D.C. Struppa, A. Vajiac and M.B. Vajiac DifferentialEquationsinMulticomplexSpaces ...................... 213 Preface At the 9th International ISAAC Congress (International Society for Analysis, its Applications, and Computations), held at the Pedagogical University of Krakow, Krakow, Poland from August 5 to August 9, 2013, one of the largest sessions was on “Clifford and Quaternionic Analysis” with around 40 speakers coming from all parts of the world: Belgium, Cech Republic, China, Finland, Germany, Israel, Italy, Mexico, Portugal, Russia, Turkey, Venezuela, Ukraine, United Kingdom and the United States. While there are official congress proceedings, the success of the session led the organizers to ask the participants to present their most recent and promising achievements in a special volume to promote the exciting field of hypercomplex analysis. This volume contains a careful selection of 15 of these papers which cover several different aspects of hypercomplex analysis going from function theory over quaternions, Clifford numbers and multicomplex numbers, operator theory, monogenic signals, to the recent field of fractional Clifford analysis. Additionally, applications to image processing, crack analysis, and the theory of elasticity are covered. All contributed papers represent the most recent achievements in the area. We hope that anybody interested in the field can find many new ideas and promising new directions in these papers. The Editors are grateful to the contributors to this volume and to the referees, for their painstaking and careful work. They also would like to thank the Peda- gogical University in Krakow for hosting the Conference and Vladimir Mityushev, in particular, as Chairman of the local organising committee. May 2014, Swanhild Bernstein Uwe K¨ahler Irene Sabadini Frank Sommen Hypercomplex Analysis: New Perspectives and Applications Trends in Mathematics, 1–18 c 2014 Springer International Publishing Switzerland Symmetries and Associated Pairs in Quaternionic Analysis Ricardo Abreu Blaya, Juan Bory Reyes, Al´ıGuzm´an Ad´an and Uwe K¨ahler Abstract. The present paper is aimed at proving necessary and sufficient con- ditions on the quaternionic-valued coefficients of a first-order linear operator to be associated to the generalized Cauchy–Riemann operator in quarternionic analysis and explicitly we give the description of all its nontrivial first-order symmetries. Mathematics Subject Classification (2010). 30G35. Keywords. Quaternionic analysis, generalized Cauchy–Riemann operator, symmetries and associated pairs. 1. Motivation and basic facts of quaternionic analysis Approaches by symmetry operators and methods based on associated pairs not only play an important role for finding explicit solutions to systems of partial dif- ferential equations, see for instance [5, 9, 10, 11, 15, 16, 17], but are also closely linked to invariance groups of operators. One of the important points is that first- order symmetries form a Lie algebra where the action of the transformation group is induced by the Lie derivatives [12]. This was used quite extensively in the past in Clifford Analysis [23, 21, 22, 3]. Furthermore, the study of first-order symmetries of the Cauchy–Riemann–Fueter operator, as well as the description of all its asso- ciated pairs, has been done recently by Y. Krasnov [9] and by T.V. Nguyen [17]. Quaternionic analysis offers a function theory related to Cauchy–Riemann– Fueter operator, which represent a generalization of classical complex analysis to This work was supported by Portuguese funds through the CIDMA – Center for Research and Development in Mathematics and Applications, and
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