Slice regular functions on real alternative algebras

Abstract

In this paper we develop a theory of slice regular functions on a real alternative algebra A. 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 A and we prove a Cauchy integral formula for slice functions of class C 1 . Keywords: Functions of a hypercomplex variable, Quaternions, Octonions, Fundamental theorem of algebra, Cauchy integral formula Math. Subj. Class: 30C15, 30G35, 32A30, 17D05 * Work partially supported by MIUR (PRIN Project "Proprietà geometriche delle varietà reali e complesse") and GNSAGA of INdAMProposition 5. Let J, K ∈ S A with J − K invertible. Every slice function f ∈ S(Ω D ) is uniquely determined by its values on the two distinct half planes C + J and C + K . In particular, taking K = −J, we have that f is uniquely determined by its values on a complex plane C J .Proof. For f ∈ S(Ω D ), let f + J be the restrictionIt is sufficient to show that the stem function F such that I(F ) = f can be recovered from the restrictions f + J , f + K . This follows from the formulawhich holds for every α + iβ ∈ D with β ≥ 0. In particular, it implies the vanishing of F 2 when β = 0. Then F 2 is determined also for β < 0 by imposing oddness w.r.t. β. Moreover, the formula f + J (α + βJ) − JF 2 (α + iβ) = F 1 (α + iβ) defines the first component F 1 as an even function on D.

Growth and distortion theorems for slice monogenic functions

“…To have a more complete insight, we refer the reader to the monographs [9,16] and the references therein. These function theories were also unified and generalized in [21] by means of the concept of slice functions on the so-called quadratic cone of a real alternative *-algebra, based on a slight modification of a well-known construction due to Fueter. The theory of slice regular functions on real alternative *-algebras is by now well-developed through a series of papers mainly due to Ghiloni and Perotti after their seminal work [21].…”

“…These function theories were also unified and generalized in [21] by means of the concept of slice functions on the so-called quadratic cone of a real alternative *-algebra, based on a slight modification of a well-known construction due to Fueter. The theory of slice regular functions on real alternative *-algebras is by now well-developed through a series of papers mainly due to Ghiloni and Perotti after their seminal work [21]. It is also well worth mentioning that this recently introduced theory of slice regular (slice monogenic) functions is significantly different from the more classical theory of regular (monogenic) functions in the sense of Cauchy-Fueter (cf.…”

On some geometric properties of slice regular functions of a quaternion variable

“…These functions are (left) slice regular according to the definition in [20] and also according to the definition in [21]. The two definitions in [20] and in [21] are different, but they give rise to the same class of functions on some particular opens sets called axially symmetric slice domains, that we will introduce in the next section.…”

“…These functions are (left) slice regular according to the definition in [20] and also according to the definition in [21]. The two definitions in [20] and in [21] are different, but they give rise to the same class of functions on some particular opens sets called axially symmetric slice domains, that we will introduce in the next section. Slice regular functions are nowadays a widely studied topic, important especially for its application to a functional calculus for quaternionic linear operators, see [12], and to Schur analysis, see [4], and [5] in which Blaschke factors are also studied.…”

Algebraic Approach to Slice Monogenic Functions

“…Also more elaborate generalisations, e.g. using real alternative algebras, have been investigated (see [17]). In other approaches to hypercomplex analysis, an important role is played by an underlying algebraic structure, namely the Lie superalgebra osp(1|2), which allows to find a representation theoretic interpretation of various function space decompositions (see [12]).…”