The Cauchy–Kovalevskaya extension theorem in Hermitian Clifford analysis

Abstract

Hermitian Clifford analysis is a higher dimensional function theory centered around the simultaneous null solutions, called Hermitian monogenic functions, of two Hermitian conjugate complex Dirac operators. As an essential step towards the construction of an orthogonal basis of Hermitian monogenic polynomials, in this paper a Cauchy-Kovalevskaya extension theorem is established for such polynomials. The minimal number of initial polynomials needed to obtain a unique Hermitian monogenic extension is determined, along with the compatibility conditions they have to satisfy. The Cauchy-Kovalevskaya extension principle then allows for a dimensional analysis of the spaces of spherical Hermitian monogenics, i.e. homogeneous Hermitian monogenic polynomials. A version of this extension theorem for specific real-analytic functions is also obtained.

Generalized Taylor series in hermitian Clifford analysis

“…Comparing this decomposition (15) with the Fischer decomposition (12), it becomes clear that it must be possible to express the embedding factors X n,r,s a,b,c,d appearing in (14), in terms of the embedding factors F n a,b,c,d appearing in the branching (3) of H a,b (C n ). The result is the following.…”

Hermitian Clifford Analysis

“…But the Cauchy data have to satisfy extra constraints in order to admit a CK-extension, and even the initial term f 00 has to. To obtain more clarity on the CK-extension problem first studied in [16,35], in the paper [36] the authors considered a subsystem of the h -monogenic system that imposes no constraints on the initial term f 00 . It is easily obtained as follows.…”

“…In other words, the CK-extension problem as discussed in [16] is part of the CK-extension problem of the more general h -submonogenic system. In [36] several new special solutions of the h -submonogenic system were constructed, involving Bessel functions, hypergeometric functions and Laguerre polynomials; these were obtained by choosing the initial term A 00 in a special way and the obtained special solutions are h -submonogenic but not h -monogenic.…”

Cauchy–Kovalevskaya Extension Theorem in Fractional Clifford Analysis

“…The CK-extension in Euclidian Clifford analysis is a direct generalization to higher dimension of the complex plane case, and can be founded in [5]. Generalizations the CK-extension to another Clifford algebra settings can be found for instance in [2,6,7,8,9,15].…”