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Fischer Decomposition and Cauchy–Kovalevskaya Extension in Fractional Clifford Analysis: The Riemann–Liouville Case

Part of: Partial differential equations Real functions Generalized function theory Higher-dimensional theory General properties

Published online by Cambridge University Press:  13 June 2016

N. Vieira
N. Vieira*
Affiliation:
CIDMA – Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal ([email protected])
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Abstract

In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations via fractional Riemann–Liouville derivatives. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers are constructed. Moreover, we establish the fractional Cauchy–Kovalevskaya extension (FCK extension) theorem for fractional monogenic functions defined on ℝd . Based on this extension principle, fractional Fueter polynomials, forming a basis of the space of fractional spherical monogenics, i.e. fractional homogeneous polynomials, are introduced. We study the connection between the FCK extension of functions of the form xPl and the classical Gegenbauer polynomials. Finally, we present an example of an FCK extension.

Keywords

fractional monogenic polynomialsFischer decompositionAlmansi decompositionCauchy–Kovalevskaya extension theoremfractional Clifford analysisfractional Dirac operator

MSC classification

Secondary: 30G35: Functions of hypercomplex variables and generalized variables 26A33: Fractional derivatives and integrals 35A10: Cauchy-Kovalevskaya theorems 30A05: Monogenic properties of complex functions (including polygenic and areolar monogenic functions) 31B05: Harmonic, subharmonic, superharmonic functions 30G20: Generalizations of Bers or Vekua type (pseudoanalytic, $p$-analytic, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 1 , February 2017 , pp. 251 - 272
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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