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Erschienen in:
Some General Properties of Operators in Morrey-Type Spaces Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

Harmonic Analysis and Hypercomplex Function Theory in Co-dimension One

verfasst von : Helmuth R. Malonek, Isabel Cação, M. Irene Falcão, Graça Tomaz

Erschienen in: Modern Methods in Operator Theory and Harmonic Analysis

Verlag: Springer International Publishing

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Abstract

Fundamentals of a function theory in co-dimension one for Clifford algebra valued functions over \({\mathbb {R}}^{n+1}\) are considered. Special attention is given to their origins in analytic properties of holomorphic functions of one and, by some duality reasons, also of several complex variables. Due to algebraic peculiarities caused by non-commutativity of the Clifford product, generalized holomorphic functions are characterized by two different but equivalent properties: on one side by local derivability (existence of a well defined derivative related to co-dimension one) and on the other side by differentiability (existence of a local approximation by linear mappings related to dimension one). As important applications, sequences of harmonic Appell polynomials are considered whose definition and explicit analytic representations rely essentially on both dual approaches.

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Vorheriges Kapitel Hardy Type Inequalities in the Category of Hausdorff Operators
Nächstes Kapitel Paraproduct in Besov–Morrey Spaces
Fußnoten
1
The inner product in \(L_2\) is given by \( (f,g)_{{\mathcal {C}\ell }_{0,n}}=\int _{B^{n+1}}\bar{f}\,g\,d\lambda ^{n+1} \), where \(\lambda ^{n+1}\) is the Lebesgue measure and \(B^{n+1}\) is the unit ball in \(\mathbb {R}^{n+1}\).
 
Literatur
1.
Stein, E.M., Weiss, G.: Generalization of the Cauchy-Riemann equations and representations of the rotation group. Am. J. Math. 90(1), 163–196 (1968)MathSciNetCrossRef
2.
Brackx, F., Delanghe, R., De Schepper, H.: Hardy spaces of solutions of generalized Riesz and Moisil-Teodorescu systems. Complex Var. Elliptic Equ. 57(7–8), 771–785 (2012)MathSciNetCrossRef
3.
Moisil, G., Teodorescu, N.: Functions holomorphes dans l’espace. Mathematica Cluj 5, 142–159 (1931)MATH
4.
Delanghe, R., Sommen, F., Souček, V.: Clifford algebra and spinor-valued functions. A function theory for the Dirac operator. Mathematics and its Applications, vol. 53. Kluwer Academic Publishers, Dordrecht (1992)CrossRef
5.
Delanghe, R., Lávička, R., Souček, V.: The Gelfand–Tsetlin bases for Hodge–de Rham systems in Euclidean spaces Math. Methods Appl. Sci. 35(7), 745–757 (2012)MathSciNetCrossRef
6.
Delanghe, R.: On regular-analytic functions with values in a Clifford Algebra. Math. Ann. 185, 91–111 (1970)MathSciNetCrossRef
7.
8.
Kravchenko, V.V.: Applied quaternionic analysis. Research and Experience in Mathematics, vol. 28, Heldermann Verlag, Lemgo (2003)
9.
Delanghe, R.: Clifford analysis: history and perspective. Comput. Methods Funct. Theory 1(1), 107–153 (2001)MathSciNetCrossRef
10.
Fueter, R.: Über Funktionen einer Quaternionenvariablen. Atti Congr. Int. Mat, Bologna (1928)MATH
11.
Malonek, H.: Rudolf Fueter and his motivation for hypercomplex function theory. Adv. Appl. Clifford Algebr. 11(S2), 219–230 (2001)MathSciNetCrossRef
12.
Fueter, R.: Functions of a hyper complex variable. Manuscript of Lecture Notes, Fall Semester 1948/49, written and supplemented by E. Bareiss, ETH Bibliothek Zürich, 318p (1950)
13.
Gürlebeck, K., Habetha, K., Sprößig, W.: Holomorphic Functions in the Plane and \(n\)-Dimensional Space. Birkhäuser Verlag, Basel (2008)MATH
14.
Gürlebeck, K., Sprößig, W.: Quaternionic analysis and elliptic boundary value problems. International Series of Numerical Mathematics, vol. 89, Birkhäuser-Verlag, Basel (1990)CrossRef
15.
Ryan, J.: Clifford analysis with generalize elliptic and quasi-elliptic functions. Appl. Anal. 13, 151–171 (1982)CrossRef
16.
Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman, Boston, London, Melbourne (1982)MATH
17.
Malonek, H.: A new hypercomplex structure of the Euclidean space \(\mathbb{R}^{n+1}\) and a concept of hypercomplex differentiability. Complex Variables, Theory Appl. 14, 25–33 (1990)MathSciNetCrossRef
18.
Malonek, H.: The concept of hypercomplex differentiability and related differential forms. In: Kühnau, R., Tutschke, W. (eds.) Studies in Complex Analysis and its Applications to Partial Differential Equations 1, vol. 256, pp. 193–202. Pitman , Longman (1991)
19.
Malonek, H.: Hypercomplex derivability—The characterization of monogenic functions in \({\mathbb{R}}^{n+1}\) by their derivative. In: Ryan, J. et al. (eds.) Clifford Algebras and their Applications in Mathematical Physics, vol. 2. Progress in Physics, vol. 19, Birkhäuser (2000)
20.
Malonek, H.R.: Selected topics in hypercomplex function theory. In: Eriksson, S.-L. (ed.) Clifford Algebras and Potential Theory. Report Series 7, University of Joensuu, pp. 111–150 (2004)
21.
Gürlebeck, K., Malonek, H.: A hypercomplex derivative of monogenic functions in \(\mathbb{R}^{n+1}\) and its applications. Complex Var. Theory Appl. 39, 199–228 (1999)MathSciNetMATH
22.
Ryan, J.: Clifford algebras in analysis and related topics. Studies in Advanced Mathematics, CRC PRESS (1996)
23.
Pompeiu, M.: Sur une classe de fonctions dúne variable complexe. Rend. Circ. Mat. Palermo 33, 108– 113 (1912), 35, 277–281 (1913)
24.
25.
Mitrea, M., Sabac, F.: Pompeiu’s integral representation formula. History and Mathematics. Rev. Roumaine Math. Pures Appl. 43, 211–226 (1998)
26.
Edwards, H.E.: Advanced Calculus: A Differential Forms Approach, 3rd edn. Birkhäuser, Boston, Basel, Berlin (1993)
27.
Malonek, H.: On the concept of holomorphy in higher dimensions—Hypercomplex differentiability and series in permutative powers. Habilitation thesis, Halle (1987)
28.
29.
Malonek, H.: Power series representation for monogenic functions in \(\mathbb{R}^{n+1}\) basead on a permutational product. Complex Var. Theory Appl. 15, 181–191 (1990)MathSciNetMATH
30.
31.
32.
Aceto, L., Malonek, H.R., Tomaz, G.: A unified matrix approach to the representation of Appell polynomials. Integral Transf. Spec. Funct. 26(6), 426–441 (2015)MathSciNetCrossRef
33.
Malonek, H.R., Falcão, M.I.: Special monogenic polynomials—properties and applications. In: Simos, T.S., Psihoyios, G., Tsitouras, C. (eds.) AIP Conference Proceedings, vol. 936, pp. 765–767 (2007)
34.
Cação, I., Falcão, M.I., Malonek, H.R.: Laguerre derivative and monogenic Laguerre polynomials: an operational approach. Math. Comput. Model. 53(5–6), 1084–1094 (2011)MathSciNetCrossRef
35.
Bock, S., Gürlebeck, K.: On a generalized Appell system and monogenic power series. Math. Methods Appl. Sci. 33(4), 394–411 (2010)MathSciNetMATH
36.
Eelbode, D.: Monogenic Appell sets as representations of the Heisenberg algebra. Adv. Appl. Clifford Algebra 22(4), 1009–1023 (2012)MathSciNetCrossRef
37.
Làvička, R.: Complete orthogonal Appell systems for spherical monogenics. Complex Anal. Oper. Theory 6, 477–489 (2012)MathSciNetCrossRef
38.
Bock, S., Gürlebeck, K., Làvička, R., Souček, V.: Gelfand-Tstelin bases for spherical monogenics in dimension 3. Rev. Mat. Iberuom. 28(4), 1165–1192 (2012)CrossRef
39.
Cação, I., Falcão, M.I., Malonek, H.R.: Matrix representations of a basic polynomial sequence in arbitrary dimension. Comput. Methods Funct. Theory 12(2), 371–391 (2012)MathSciNetCrossRef
40.
Cação, I., Falcão, M.I., Malonek, H.R.: Three term recurrence relations for systems of Clifford algebra-valued polynomial sequences. Adv. Appl. Clifford Algebras 27, 71–85 (2017)MathSciNetCrossRef
41.
Aceto, L., Malonek, H.R., Tomaz, G.: Matrix approach to hypercomplex Appell polynomials. Appl. Numer. Math. 116, 2–9 (2017)MathSciNetCrossRef
42.
43.
Cruz, C., Falcão, M.I., Malonek, H.R.: Monogenic pseudo-complex power functions and their applications. Math. Methods Appl. Sci. 37, 1723–1735 (2014)MathSciNetCrossRef
44.
45.
Cação, I., Falcão, M.I., Malonek, H.R.: Hypercomplex polynomials, Vietoris’ rational numbers and a related integer numbers sequence. Complex Anal. Oper. Theory 11, 1059–1076 (2017)MathSciNetCrossRef
46.
Cação, I., Falcão, M.I., Malonek, H.R., Tomaz, G.: Combinatorial identities associated with a multidimensional polynomial sequence. J. Integer Seq. 21(7), 18.7.4 (2018)
Metadaten
Titel
Harmonic Analysis and Hypercomplex Function Theory in Co-dimension One
verfasst von
Helmuth R. Malonek
Isabel Cação
M. Irene Falcão
Graça Tomaz
Copyright-Jahr
2019
Buch
Modern Methods in Operator Theory and Harmonic Analysis

Print ISBN: 978-3-030-26747-6

Electronic ISBN: 978-3-030-26748-3

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-26748-3_7