Cauchy Kovalevskaya Extension Research Articles

Szegö-Radon transform for hypermonogenic functions

In this paper, we study a refinement of the Szegö-Radon transform in the hypermonogenic setting. Hypermonogenic functions form a subclass of monogenic functions arising in the study of a modified Dirac operator, which allows for weaker symmetries and also has a strong connection to the hyperbolic metric. In particular, we construct a projection operator from a module of hypermonogenic functions in Rp+q onto a suitable submodule of plane waves parameterized by a vector on the unit sphere of Rq. Moreover, we study the interaction of this Szegö-Radon transform with the generalized Cauchy-Kovalevskaya extension operator. Finally, we develop a reconstruction (inversion) method for this transform.

Generalized Cauchy–Kovalevskaya extension and plane wave decompositions in superspace

The aim of this paper is to obtain a generalized CK-extension theorem in superspace for the biaxial Dirac operator $$\partial _{\mathbf{x}} +\partial _{\mathbf{y}}$$ . In the classical commuting case, this result can be written as a power series of Bessel type of certain differential operators acting on a single initial function. In the superspace setting, novel structures appear in the cases of negative even superdimensions. In these cases, the CK-extension depends on two initial functions on which two power series of differential operators act. These series are not only of Bessel type but they give rise to an additional structure in terms of Appell polynomials. This pattern also is present in the structure of the Pizzetti formula, which describes integration over the supersphere in terms of differential operators. We make this relation explicit by studying the decomposition of the generalized CK-extension into plane waves integrated over the supersphere. Moreover, these results are applied to obtain a decomposition of the Cauchy kernel in superspace into monogenic plane waves, which shall be useful for inverting the super Radon transform.

The Monogenic Cauchy–Kovalevskaya Extension in Two Vector Variables

The Cauchy–Kovalevskaya extension for polynomials in several variables is a crucial result in Clifford analysis which describes theoretically how to construct simplicial monogenics in m dimensions starting from certain polynomial spaces in \(m-1\) dimensions, hereby using the so-called branching rules. In [13] it was shown that the Cauchy–Kovalevskaya extension map is an isomorphism of the kernel of the wedge operator onto the space of spherical monogenics in two variables. The aim of this paper is to show that this extension maps simplicial polynomials of the kernel of the wedge operator just onto the simplicial monogenics.

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

AbstractIn 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.

A Compact Cauchy-Kovalevskaya Extension Formula in Discrete Clifford Analysis

We prove a compact expression for the Cauchy-Kovalevskaya extension in the setting of discrete Clifford analysis; it seamlessly simplifies to the well-know exponential formula in the continuous setting when the discrete step size tends to zero.

Cauchy–Kovalevskaya Extension Theorem in Fractional Clifford Analysis

In this paper, we establish the fractional Cauchy–Kovalevskaya extension (\(\textit{FCK}\)-extension) theorem for fractional monogenic functions defined on \(\mathbb {R}^d\). 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 studied the connection between the \(\textit{FCK}\)-extension of functions of the form \(x^\alpha P_l\) and the classical Gegenbauer polynomials. Finally we present two examples of \(\textit{FCK}\)-extension.

Fueter polynomials in discrete Clifford analysis

Discrete Clifford analysis is a higher dimensional discrete function theory, based on skew Weyl relations. The basic notions are discrete monogenic functions, i.e. Clifford algebra valued functions in the kernel of a discrete Dirac operator. In this paper, we introduce the discrete Fueter polynomials, which form a basis of the space of discrete spherical monogenics, i.e. discrete monogenic, homogeneous polynomials. Their definition is based on a Cauchy–Kovalevskaya extension principle. We present the explicit construction for this discrete Fueter basis, in arbitrary dimension m and for arbitrary homogeneity degree k.

The Cauchy-Kovalevskaya extension theorem in discrete Clifford analysis

Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic functions, of a discrete Dirac operator, i.e. a first order, Clifford vector valued difference operator. In this paper, we establish a Cauchy-Kovalevskaya extension theorem for discrete monogenic functions defined on the standard $Z^m$ grid. Based on this extension principle, discrete Fueter polynomials, forming a basis of the space of discrete spherical monogenics, i.e. homogeneous discrete monogenic polynomials, are introduced. As an illustrative example we moreover explicitly construct the Cauchy-Kovalevskaya extension of the discrete delta function. These results are then generalized for a grid with variable mesh width $h$.

The Cauchy–Kovalevskaya extension theorem in Hermitian Clifford analysis

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.