Universidade do Minho DMA Departamento de Matemática e Aplicações Universidade do Minho Campus de Gualtar 4710-057 Braga Portugal www.math.uminho.pt Universidade do Minho Escola de Ciências Departamento de Matemática e Aplicações 3D-Mappings by Means of Monogenic Functions and their Approximation H.R. Malonek a M.I. Falc˜ ao b a Departamento de Matem´ atica, Universidade de Aveiro, Portugal b Departamento de Matem´ atica e Aplica¸ c˜ oes, Universidade do Minho, Portugal Information Keywords: Clifford Analysis, monogenic functions, 3D-mappings. Original publication: Math. Methods Appl. Sci. 33 (2010), no. 4, 423–430 DOI: 10.1002/mma.1211 www.interscience.wiley.com Abstract We consider quasi-conformal 3D-mappings realized by hyper- complex differentiable (monogenic) functions and their poly- nomial approximation. Main tools are the series development of monogenic functions in terms of hypercomplex variables and the generalization of L. V. Kantorovich’s approach for approximating conformal mappings by powers of a small pa- rameter. 1 Introduction In contrast to the planar case, in R n , with n ≥ 3, the set of conformal mappings is only the set of M¨ obius transformations (due to Liouville’s theorem [9]). The difficulties in characterizing M¨ obius transformations in R 4 by some differentiability property have been studied in detail in [8]. In the case of R 4 the application of quaternions is natural, a fact that has already been noticed in [15] and [3], for instance. But the theory of generalized holomorphic functions developed on the basis of Clifford algebras (with quaternions as a special case, cf. [3]; for historical reasons they are also called monogenic functions, cf. [2]) does not cover the set of M¨ obius transformations in R n if n ≥ 3. M¨ obius transformations are not monogenic and therefore monogenic functions are not directly related to conformal mappings in R n , n ≥ 3. Here one can only expect that monogenic functions realize quasi-conformal mappings. Obviously, such a situation has originated many questions concerning the extension of theoretical and practical conformal mapping methods in C to higher dimensions in the setting of Clifford Analysis (see [12] for a special approach). Note that, in this setting, contrary to the case of several complex variables there are no restrictions on the real dimension being even or odd. This implies that the real 3-dimensional Euclidean space, the most important space for concrete applications, can be subject to a treatment similar to the complex one. Some results based on the application of Bergman’s reproducing kernel method (BKM) in the Clifford setting are described in [1]. Our goal is to present in this case study a different from BKM approach. In fact, it is an extension of ideas of L. V. Kantorovich (c.f. [7] and [4]) to the 3-dimensional case. As usual, we identify each element x =(x 0 ,x 1 ,x 2 ) ∈ R 3 with the paravector (sometimes also called reduced quaternion) z = x 0 + x 1 e 1 + x 2 e 2 . For C 1 (Ω, R 3 ) define the generalized Cauchy-Riemann operator D = ∂ ∂x0 + e 1 ∂ ∂x1 + e 2 ∂ ∂x2 .