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The inverse Fueter mapping theorem

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Abstract
In a recent paper the authors have shown how to give an integral representation of the Fueter mapping theorem using the Cauchy formula for slice monogenic functions. Specifically, given a slice monogenic function f of the form f = alpha + (omega) under bar beta (where alpha, beta satisfy the Cauchy-Riemann equations) we represent in integral form the axially monogenic function f = A + (omega) under barB (where A, B satisfy the Vekua's system) given by f(x) = Delta n-1/2 f (x) where Delta is the Laplace operator in dimension n + 1. In this paper we solve the inverse problem: given an axially monogenic function f determine a slice monogenic function f (called Fueter's primitive of f) such that f = Delta n-1/2 f (x). We prove an integral representation theorem for f in terms of f which we call the inverse Fueter mapping theorem (in integral form). Such a result is obtained also for regular functions of a quaternionic variable of axial type. The solution f of the equation Delta n-1/2 f(x) = f(x) in the Clifford analysis setting, i.e. the inversion of the classical Fueter mapping theorem, is new in the literature and has some consequences that are now under investigation.
Keywords
Fueter's primitive, Fueter mapping theorem in integral form, slice monogenic functions, axially monogenic function, Vekua's system, Cauchy-Riemann equations, inverse Fueter mapping theorem in integral form, SLICE MONOGENIC FUNCTIONS, FUNCTIONAL-CALCULUS, NONCOMMUTING OPERATORS, REGULAR FUNCTIONS, CONSEQUENCES, FORMULA

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Chicago
Colombo, Fabrizio, Irene Sabadini, and Franciscus Sommen. 2011. “The Inverse Fueter Mapping Theorem.” Ommunications on Pure and Applied Analysis 10 (4): 1165–1181.
APA
Colombo, F., Sabadini, I., & Sommen, F. (2011). The inverse Fueter mapping theorem. OMMUNICATIONS ON PURE AND APPLIED ANALYSIS, 10(4), 1165–1181.
Vancouver
1.
Colombo F, Sabadini I, Sommen F. The inverse Fueter mapping theorem. OMMUNICATIONS ON PURE AND APPLIED ANALYSIS. 2011;10(4):1165–81.
MLA
Colombo, Fabrizio, Irene Sabadini, and Franciscus Sommen. “The Inverse Fueter Mapping Theorem.” OMMUNICATIONS ON PURE AND APPLIED ANALYSIS 10.4 (2011): 1165–1181. Print.
@article{1938075,
  abstract     = {In a recent paper the authors have shown how to give an integral representation of the Fueter mapping theorem using the Cauchy formula for slice monogenic functions. Specifically, given a slice monogenic function f of the form f = alpha + (omega) under bar beta (where alpha, beta satisfy the Cauchy-Riemann equations) we represent in integral form the axially monogenic function f = A + (omega) under barB (where A, B satisfy the Vekua's system) given by f(x) = Delta n-1/2 f (x) where Delta is the Laplace operator in dimension n + 1. In this paper we solve the inverse problem: given an axially monogenic function f determine a slice monogenic function f (called Fueter's primitive of f) such that f = Delta n-1/2 f (x). We prove an integral representation theorem for f in terms of f which we call the inverse Fueter mapping theorem (in integral form). Such a result is obtained also for regular functions of a quaternionic variable of axial type. The solution f of the equation Delta n-1/2 f(x) = f(x) in the Clifford analysis setting, i.e. the inversion of the classical Fueter mapping theorem, is new in the literature and has some consequences that are now under investigation.},
  author       = {Colombo, Fabrizio and Sabadini, Irene and Sommen, Franciscus},
  issn         = {1534-0392},
  journal      = {OMMUNICATIONS ON PURE AND APPLIED ANALYSIS},
  language     = {eng},
  number       = {4},
  pages        = {1165--1181},
  title        = {The inverse Fueter mapping theorem},
  url          = {http://dx.doi.org/10.3934/cpaa.2011.10.1165},
  volume       = {10},
  year         = {2011},
}

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