Fueter's Theorem Research Articles

On axially rational regular functions and Schur analysis in the Clifford-Appell setting

In this paper we start the study of Schur analysis for Cauchy–Fueter regular quaternionic-valued functions, i.e. null solutions of the Cauchy–Fueter operator in R4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^4$$\\end{document}. The novelty of the approach developed in this paper is that we consider axially regular functions, i.e. functions spanned by the so-called Clifford-Appell polynomials. This type of functions arises naturally from two well-known extension results in hypercomplex analysis: the Fueter mapping theorem and the generalized Cauchy–Kovalevskaya (GCK) extension. These results allow one to obtain axially regular functions starting from analytic functions of one real or complex variable. Precisely, in the Fueter theorem two operators play a role. The first one is the so-called slice operator, which extends holomorphic functions of one complex variable to slice hyperholomorphic functions of a quaternionic variable. The second operator is the Laplace operator in four real variables, that maps slice hyperholomorphic functions to axially regular functions. On the other hand, the generalized CK-extension gives a characterization of axially regular functions in terms of their restriction to the real line. In this paper we use these two extensions to define two notions of rational function in the regular setting. For our purposes, the notion coming from the generalized CK-extension is the most suitable. Our results allow to consider the Hardy space, Schur multipliers and their relation with realizations in the framework of Clifford-Appell polynomials. We also introduce two notions of regular Blaschke factors, through the Fueter theorem and the generalized CK-extension.

Partial slice regularity and Fueter's theorem in several quaternionic variables

Abstract We extend some definitions and give new results about the theory of slice analysis in several quaternionic variables. The sets of slice functions that are slice, slice regular, and circular with respect to given variables are characterized. We introduce new notions of partial spherical value and derivative for functions of several variables that extend those of one variable. We recover some of their properties as circularity, harmonicity, some relations with differential operators, and a Leibniz rule with respect to the slice product as well as studying their behavior in the context of several variables. Then, we prove our main result, which is a generalization of Fueter’s theorem for slice regular functions in several variables. This extends the link between slice regular and axially monogenic functions well known in the one variable context.

A polyanalytic functional calculus of order 2 on the 𝑆-spectrum

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solution of the Cauchy-Riemann operator inR4\mathbb {R}^4, denoted byD\mathcal {D}. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operatorΔ\Deltain four real variables to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e.Δ=D¯D\Delta = \mathcal {\overline {D}} \mathcal {D}to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions ofD2\mathcal {D}^2. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on theSS-spectrum.

Implementing zonal harmonics with the Fueter principle

By exploiting the Fueter theorem, we give new formulas to compute zonal harmonic functions in any dimension. We first give a representation of them as a result of a suitable ladder operator acting on a constant function. Then, inspired by recent work of A. Perotti, using techniques from slice regularity, we derive explicit expressions for zonal harmonics starting from the 2 and 3 dimensional cases. It turns out that all zonal harmonics in any dimension are related to the real part of powers of the standard Hermitian product in C. At the end we compare formulas, obtaining interesting equalities involving the real part of positive and negative powers of the standard Hermitian product.In the two appendices we show how our computations are optimal compared to direct ones.

The Symplectic Fueter–Sce Theorem

In this paper we present a symplectic analogue of the Fueter theorem. This allows the construction of special (polynomial) solutions for the symplectic Dirac operator $D_s$, which is defined as the first-order $\mathfrak{sp}(2n)$-invariant differential operator acting on functions on ${\mathbb R}^{2n}$ taking values in the metaplectic spinor representation.

Higher spin generalisation of Fueter's theorem

In this paper, we generalize Fueter's theorem to the higher spin setting. To do so, we consider an alternative proof for the celebrated theorem that uses the Fischer decomposition. This decomposition is then extended to spaces of polynomials that depend on wedge variables, after which we can finish the proof of our higher spin Fueter theorem.

On the inversion of Fueter’s theorem

The well known Fueter theorem allows to construct quaternionic regular functions or monogenic functions with values in a Clifford algebra defined on open sets of Euclidean space R n + 1 , starting from a holomorphic function in one complex variable or, more in general, from a slice hyperholomorphic function. Recently, the inversion of this theorem has been obtained for odd values of the dimension n . The present work extends the result to all dimensions n by using the Fourier multiplier method. More precisely, we show that for any axially monogenic function f defined in a suitable open set in R n + 1 , where n is a positive integer, we can find a slice hyperholomorphic function f → such that f = Δ ( n − 1 ) / 2 f → . Both the even and the odd dimensions are treated with the same, viz., the Fourier multiplier, method. For the odd dimensional cases the result obtained by the Fourier multiplier method coincides with the existing result obtained through the pointwise differential method.

Fueter's theorem in discrete Clifford analysis

In this paper, we discretize techniques for the construction of axially monogenic functions to the setting of discrete Clifford analysis. Wherefore, we work in the discrete Hermitian Clifford setting, where each basis vector ej is split into a forward and backward basis vector: . We prove a discrete version of Fueter's theorem in odd dimension by showing that for a discrete monogenic function f(ξ0,ξ1) left‐monogenic in two variables ξ0 and ξ1 and for a left‐monogenic Pk(ξ), the m‐dimensional function is in itself left monogenic, that is, a discrete function in the kernel of the discrete Dirac operator. Closely related, we consider a Vekua‐type system for the construction of axially monogenic functions. We consider some explicit examples: the discrete axial‐exponential functions and the discrete Clifford–Hermite polynomials. Copyright © 2015 John Wiley & Sons, Ltd.

Biaxial monogenic functions from Funk‐Hecke's formula combined with Fueter's theorem

Funk‐Hecke's formula allows a passage from plane waves to radially invariant functions. It may be adapted to transform axial monogenics into biaxial monogenics that are monogenic functions invariant under the product group SO(p)× SO(q). Fueter's theorem transforms holomorphic functions in the plane into axial monogenics, so that by combining both results, we obtain a method to construct biaxial monogenics from holomorphic functions.

The biaxial Fueter theorem

This paper deals with a special type of solutions for the Dirac operator on ℝ m , which can be obtained through a biaxial generalisation of the classical Fueter Theorem. This is a result which allows to generate zonal solutions for the Dirac equation starting from arbitrary holomorphic functions in the complex plane. Invoking operator identities for Jacobi polynomials, it is shown how this procedure can be extended to more general splittings than the one usually considered in the literature.

A generalization of Fueter's theorem in Dunkl–Clifford analysis

In this article, we first suggest an alternative approach to extend the original Fueter's theorem in Dunkl–Clifford analysis to a version of the higher order case. We then use this result to prove a generalized version of Fueter's theorem with an extra homogeneous Dunkl-monogenic polynomial P n (x 0, x ) in instead of the classical factor P n ( x ) in ℝ d .

Gegenbauer polynomials and the Fueter theorem

The Fueter theorem states that regular (resp. monogenic) functions in quaternionic (resp. Clifford) analysis can be constructed from holomorphic functions in the complex plane, hereby using a combination of a formal substitution and the action of an appropriate power of the Laplace operator. In this paper we interpret this theorem on the level of representation theory, as an intertwining map between certain -modules.

Fueter's theorem: The saga continues

In this paper is extended the original theorem by Fueter–Sce (assigning an R 0 , m -valued monogenic function to a C -valued holomorphic function) to the higher order case. We use this result to prove Fueter's theorem with an extra monogenic factor P k ( x 0 , x ̲ ) .

Fueter's theorem and its generalizations in Dunkl–Clifford analysis

In this paper, we give a construction of Dunkl monogenic and Dunkl harmonic functions starting from holomorphic functions in the plane. This construction has the advantage of not needing Dunkl's intertwining operator or Dunkl spherical harmonics. To this end we study Vekua-type systems and prove a version of Fueter's theorem in the case of finite reflection groups. Important examples, such as a Dunkl monogenic Gaussian distribution or a Cauchy kernel, will be given at the end.

Pointwise convergence for expansions in spherical monogenics

We offer a new approach to deal with the pointwise convergence of Fourier-Laplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter's theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson's theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson's theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.

Clifford algebra approach to pointwise convergence of Fourier series on spheres

We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter's Theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini's type pointwise convergence theorem are proved.

An alternative proof of Fueter's theorem§

In this article we establish an alternative proof of the generalized Fueter method presented in a former paper [Qian, T. and Sommen, F., 2003, Deriving harmonic functions in higher dimensional spaces. Zeitschrift fur Analysis und ihre Anwendungen, 22(2), 275–288] leading to the construction of special harmonic and monogenic functions in higher dimensions. At the same time, we also obtain a generalization of this result. §Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

Deriving Harmonic Functions in Higher Dimensional Spaces

For a harmonic function, by replacing its variables with norms of vectors in some multi-dimensional spaces, we may induce a new function in a higher dimensional space. We show that, after applying to it a certain power of the Laplacian, we obtain a new harmonic function in the higher dimensional space. We show that Poisson and Cauchy kernels and Newton potentials, and even heat kernels are all deducible using this method based on their forms in the lowest dimensional spaces. Fueter's theorem and its generalizations are deducible as well from our results. The latter has been used to singular integral and Fourier multiplier theory on the unit spheres and their Lipschitz perturbations of higher dimensional Euclidean spaces.