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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
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Contents
Preface .................................................................. vii
R. Abreu Blaya, J. Bory Reyes, A. Guzma´n 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-space and Krein–Langer Factorization .......................... 19
D. Alpay, F. Colombo, I. Sabadini and G. Salomon
The Fock Space in the Slice Hyperholomorphic Setting .............. 43
E. Ariza and A. Di Teodoro
Multi Mq-monogenic Function in Different Dimension ............... 61
S. Bernstein
The Fractional Monogenic Signal .................................... 75
L.J. Carmona L., L.F. Res´endis O. and L.M. Tovar S.
Weighted Bergman Spaces .......................................... 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. Gu¨rlebeck and D. Legatiuk
On the Continuous Coupling of Finite Elements
with Holomorphic Basis Functions .................................. 167
vi Contents
K. Gu¨rlebeck and H. Manh Nguyen
On ψ-hyperholomorphic Functions and
a Decomposition of Harmonics ...................................... 181
U. Ka¨hler and N. Vieira
Fractional Clifford Analysis ......................................... 191
Y. Krasnov
Spectral Properties of Differential Equations in Clifford Algebras .... 203
D.C. Struppa, A. Vajiac and M.B. Vajiac
Differential Equations in Multicomplex Spaces ...................... 213
Preface
At the 9th International ISAAC Congress (International Society for Analysis, its
Applications, and Computations), held at the Pedagogical University of Krakow,
Krakow,Polandfrom 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,UnitedKingdomand
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
coverseveraldifferentaspectsofhypercomplexanalysisgoingfromfunctiontheory
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.
TheEditorsaregratefultothecontributorstothisvolumeandtothereferees,
for their painstaking and careful work. They also would like to thank the Peda-
gogicalUniversityinKrakowfor hostingthe Conference andVladimir Mityushev,
in particular, as Chairman of the local organising committee.
May 2014, Swanhild Bernstein
Uwe K¨ahler
Irene Sabadini
FrankSommen
HypercomplexAnalysis:NewPerspectives andApplications
TrendsinMathematics,1–18
(cid:2)c 2014SpringerInternational PublishingSwitzerland
Symmetries and Associated Pairs
in Quaternionic Analysis
Ricardo Abreu Blaya, Juan Bory Reyes, Al´ı Guzma´n Ad´an
and Uwe Ka¨hler
Abstract. Thepresentpaperisaimedatprovingnecessaryandsufficientcon-
ditions on thequaternionic-valued coefficients of a first-order linear operator
tobeassociatedtothegeneralizedCauchy–Riemannoperatorinquarternionic
analysis and explicitly we give the description of all its nontrivial first-order
symmetries.
Mathematics SubjectClassification (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
linkedto invariancegroupsofoperators.One ofthe importantpoints is thatfirst-
ordersymmetriesformaLiealgebrawherethe actionofthetransformationgroup
isinducedbytheLiederivatives[12].Thiswasusedquiteextensivelyinthepastin
Clifford Analysis [23, 21, 22, 3]. Furthermore, the study of first-order symmetries
of the Cauchy–Riemann–Fueteroperator,as well as the descriptionof 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 the Portuguese Foundation for Sci-
ence and Technology (“FCT – Funda¸ca˜o para aCiˆencia e a Tecnologia”), withinproject PEst-
OE/MAT/UI4106/2014.
2 R. Abreu Blaya, J. Bory Reyes, A. Guzma´n Ada´n and U. K¨ahler
higher-dimensional Euclidean space. For a more detailed account on this matter
along more traditional lines we refer the reader to [2, 7, 8].
StartingfromadefinitionofageneralizedCauchy–Riemannoperator(related
to some fixed orthonormal basis in R4) proposed by Vasilevski and Shapiro ([24])
quaternionic analysis shows important advantages in the effort of solving partial
differential equations inthe hypercomplexframework.Their approachis basedon
thenotionofastructuralsetψk,representingageneralorthonormalframe.While
the obtained function theory is in most aspects the same as for the generalized
Cauchy–Riemann operator with respect to the standard basis this notion is the
starting pointfor addressingproblemswhich arisewhen more thanone structural
set is involved. Such cases arise, for example, when one studies derivatives of
monogenic functions with respect to a different orthonormal frame. While in the
quaternionicsettingMitelman/Shapiro[13]showedthattheconjugategeneralized
Cauchy–Riemannoperatorcanplaytheroleofaψ-hyperholomorphicderivative,it
cannotseparatealldirections(whichlaterledtothenotionofahyperholomorphic
constant), which means that to recover all partial derivatives like in the complex
case one would need three differential operators where each of them is obtained
by conjugating only one of the elements of the structural set.
Another example is the study of Beltrami equations, in particular the study
of the monogenic part of its solution. Hereby, one major problem arises. While
one structural set can be transformed into another by an orthogonal transfor-
mation, in general this transformation will not be a rotation. But a generalized
Cauchy–Riemannoperator itself is only invariant under rotations, not under gen-
eral orthogonal transformations. This leads to the notion of left- (right-) equiv-
alent structural sets as the subclass of structural sets which can be transformed
into each other by an appropriate rotation. This leads to such interesting proper-
ties thatwhileforleft- (right-)equivalentstructuralsetsthe correspondingsetsof
ψ-hyperholomorphicfunctions are coinciding that is not anymore true for general
structural sets.
Additionally, while inthe frameworkofthe quaternionicalgebraitis natural
to represent rotations as multiplications of quaternions, this is not possible, for
example, in case of a reflection. Therefore, the treatment of general orthogonal
transformationsrequirestheembeddingofthequaternionsinahigher-dimensional
algebra. This can be done either by considering quaternions as 4×4-dimensional
real matrices and applying a general orthogonal transformation to this matrix or
by embedding the quaternions into a higher-dimensional algebra with signature
(4,4). In both cases the resulting algebra of endomorphisms has to be isomorphic
tothefullmatrixalgebraof4×4-matrices,notjustthesub-algebraofquaternions.
The presentpaper is aimedat provingnecessaryandsufficient conditions on
thequaternionic-valuedcoefficientsofafirst-orderlinearoperatortobeassociated
to the generalized Cauchy–Riemann operator in quarternionic analysis and we
explicitly describe all its nontrivial first-ordersymmetries. This is done at first by
linking with matrix description of our operator. In the last section we will show
that the traditional approach by Sommen/van Acker [21, 22] via identifying the
