161 research outputs found
Holomorphic functions and regular quaternionic functions on the hyperkähler space H
Let H be the space of quaternions, with its standard hypercomplex structure. Let be the module of \emph{-regular} functions on . For every , , contains the space of holomorphic functions w.r.t. the complex structure induced by . We prove the existence, on any bounded domain , of -regular functions that are not -holomorphic for any . Our starting point is a result of Chen and Li concerning maps between hyperk\"ahler manifolds, where a similar result is obtained for a less restricted class of quaternionic maps. We give a criterion, based on the energy-minimizing property of holomorphic maps, that distinguishes -holomorphic functions among -regular functions
An application of biregularity to quaternionic Lagrange interpolation
We revisit the concept of totally analytic variable of one quaternionic variable introduced by Delanghe \cite Delanghe} and its application to Lagrange interpolation by G\"uerlebeck and Spr\"ossig \cite{GS}. We consider left-regular functions in the kernel of the Cauchy-Riemann operator For every imaginary unit p\in {\Sp}^2, let {\CC}_p=\langle 1,p\rangle\simeq {\CC} and let be the corresponding complex structure on {\HH}. We identify totally regular variables with real--affine holomorphic functions from ({\HH},J_p) to ({\CC}_p,L_p), where is the complex structure defined by left multiplication by . We then show that every --biholomorphic map, which is always a biregular function, gives rise to a Lagrange interpolation formula at any set of distinct points in {\HH}. Publisher version at: http://link.aip.org/link/?APCPCS/1048/691/
A four dimensional Bernstein Theorem
We prove a four dimensional version of the Bernstein Theorem, with complex
polynomials being replaced by quaternionic polynomials. We deduce from the
theorem a quaternionic Bernstein's inequality and give a formulation of this
last result in terms of four-dimensional zonal harmonics and Gegenbauer
polynomials
The quaternionic Gauss-Lucas Theorem
The classic Gauss-Lucas Theorem for complex polynomials of degree has
a natural reformulation over quaternions, obtained via rotation around the real
axis. We prove that such a reformulation is true only for . We present a
new quaternionic version of the Gauss-Lucas Theorem valid for all ,
together with some consequences.Comment: 7 pages, 1 figure. Remarks added in section 3. Proposition 14 added
with complete proo
Slice regular functions on real alternative algebras
In this paper we develop a theory of slice regular functions on a real alternative algebra . Our approach is based on a well-known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in and we prove a Cauchy integral formula for slice functions of class
- …