The Weierstrass Factorization Theorem for Slice Regular Functions Over the Quaternions

Ann Glob Anal Geom DOI 10.1007/s10455-011-9266-0 ORIGINAL PAPER The Weierstrass factorization theorem for slice regular functions over the quaternions Graziano Gentili · Irene Vignozzi Received: 8 November 2010 / Accepted: 5 April 2011 © Springer Science+Business Media B.V. 2011 Abstract The class of slice regular functions of a quaternionic variable has been recently introduced and is intensively studied, as a quaternionic analogue of the class of holomorphic functions. Unlike other classes of quaternionic functions, this one contains natural quaternionic polynomials and power series. Its study has already produced a rather rich theory having steady foundations and interesting applications. The main purpose of this article is to prove a Weierstrass factorization theorem for slice regular functions. This result holds in a formula- tion that reflects the peculiarities of the quaternionic setting and the structure of the zero set of such functions. Some preliminary material that we need to prove has its own independent interest, like the study of a quaternionic logarithm and the convergence of infinite products of quaternionic functions. Keywords Functions of a quaternionic variable · Weierstrass factorization theorem · Zeros of hyperholomorphic functions Mathematics Subject Classification (2000) 30G35 · 30C15 · 30B10 1 Introduction The well-known theory of Fueter regular functions [9,10,25] is the first and most celebrated candidate among a few others, in the search for a quaternionic analogue of the theory of holomorphic functions of one complex variable. This well-developed theory [3,19–21] inspired also the setting of Clifford algebras (see, e.g. [2]). Recently, Gentili and Struppa proposed a different approach, which led to a new notion of holomorphicity (called slice regularity) for quaternion-valued functions of a quaternionic variable [15,16]. Unlike Fueter’s, this G. Gentili (B) · I. Vignozzi Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy e-mail: [email protected]fi.it I. Vignozzi e-mail: [email protected]fi.it 123 Ann Glob Anal Geom theory includes the polynomials and the power series of the quaternionic variable q of the n ∈ H type n≥0 q an, with coefficients an . Furthermore, the analogs (sometime peculiarly different) of many of the fundamental properties of holomorphic functions of one complex variable can be proven in this new setting, like the Cauchy and Pompeiu representation formu- las and Cauchy inequalities, the maximum (and minimum) modulus principle, the identity principle, the open mapping theorem, the Morera theorem, the power and Laurent series expansion, the Runge approximation theorem, to cite only some of the most significant (see [4–7,12–16,24]). In fact, the theory of slice regular functions is already rather rich and well established on steady foundations, and appears to be of fundamental importance to construct a functional calculus in non commutative settings [8]. Let H denote the skew field of quaternions. Its elements are of the form q = x0 + 2 2 2 ix1 + jx2 + kx3 where the xl are real, and i, j, k are such that i = j = k =−1, ij =−ji = k, jk =−kj = i, ki =−ik = j. We set Re(q) = x0,Im(q) = ix1 + jx2 +kx3, | |= 2 + 2 + 2 + 2 ( ) ( ) | | q x0 x1 x2 x3 and call Re q ,Imq and q the real part,theimaginary part and the module of q, respectively. The conjugate√of the√ quaternion q,definedasq¯ = Re(q)−Im(q) = x0 −ix1 − jx2 −kx3, satisfies |q|= qq¯ = qq¯ and allows the definition ¯ inverse q = q−1 = q . S of the of any element 0as |q|2 If is the unit sphere of purely imaginary S ={ ∈ H : 2 =− }={ = + + : 2 + 2 + 2 = } quaternions, i.e., q q 1 q ix1 jx2 kx3 x1 x2 x3 1 , then every quaternion q which is not real (i.e. with Im(q) = 0) can be written as q = x + Iy = ( ), =| ( )| = Im(q) ∈ S ∈ S for x Re q y Im q and I |Im(q)| . Moreover, using the same I , we can write q =|q|eϑ I ,forsomeϑ ∈ R. We can now state the following definition, see [4]. Definition 1.1 Let be a domain in H. A function f : → H is said to be slice regular if, for every I ∈ S, its restriction fI to the complex line L I = R + RI passing through the origin and containing 1 and I has continuous partial derivatives and satisfies 1 ∂ ∂ ∂ I f (x + yI) := + I fI (x + yI) = 0, 2 ∂x ∂y in I = ∩ L I . In what follows, we will simply call regular a slice regular function. If is a generic domain of H, then there are examples of regular functions defined on which are not even continuous, see e.g. [4]. In [5], and in this same article [4], a class of pathology-preventing domains of definition for regular functions were introduced, as follows: Definition 1.2 Let be a domain in H, intersecting the real axis. If I = ∩ L I is a domain in L I C for all I ∈ S then we say that is a slice domain. As it is well known, the domains of holomorphy are the most natural domains of definition for holomorphic functions of a complex variable. The same can be said, in the setting of quaternionic regular functions, for slice domains which have the following additional property: Definition 1.3 A subset C of H is axially symmetric if, for all x + yI ∈ C with x, y ∈ R, I ∈ S, the whole set x + yS ={x + yJ : J ∈ S} is contained in C. For the sake of simplicity, we will call such a C a symmetric set. As it is proved in [4], any regular function defined on a slice domain can be (uniquely) extended to the smallest symmetric slice domain which contains , and for this reason we will always consider regular functions defined on symmetric slice domains. 123 Ann Glob Anal Geom One of the interesting, basic features of regular functions on slice domains is the structure of their zero set: it consists of isolated points (zeros) and isolated spheres of type x + yS, with x, y ∈ R (spherical zeros). The fact that pointwise product does not preserve regularity led to the definition of a regular product (the ∗-product) which maintains regularity and allows the study of a factorization for quaternionic regular polynomials, as well as the construction of different notions of multiplicity for their roots (and for the zeros of regular functions), [11,17,18,23]. This article is devoted to the study of a quaternionic version of the well-known Weierst- rass factorization theorem, which will be valid for entire regular functions, i.e. for regular functions defined on the whole of H. The main result that we obtain is in fact the following. Theorem 1.4 (Weierstrass factorization theorem for regular functions) Let f be an entire regular function. Let: m ∈ N be the multiplicity of 0 as zero of f , {bn}n∈N ⊆ R \{0} be the sequence of the (non zero) real zeros of f , {Sn = xn + ynS}n∈N be the sequence of the spherical zeros of f ,and{an}n∈N ⊆ H \ R be the sequence of the non real zeros of f with isolated multiplicity greater then zero. If all the zeros listed above are repeated according to their multiplicities, then there exists a never vanishing, entire regular function h and, for all ∈ N ∈ δ ∈ = ( ) +| ( )|S , , ∈ N n ,thereexistcn Sn, n San Re an Im an , rn n mn such that f (q) = qm R(q) S(q) A(q) ∗ h(q) where, with obvious notations for the infinite ∗-product, ∞ − − qb−1+ 1 q2b−2+···+ 1 qrn b rn R( ) = ( − 1) n 2 n rn n , q 1 qbn e n=0 ∞ ( 2) ( n ) 2 ( ) 2Re(cn ) 1 2 2Re cn 1 n 2Re cn q 2qRe cn q + q +···+ q S(q) = − + 1 e |cn |2 2 |cn |4 n |cn |2 n , |c |2 |c |2 n=0 n n ∞ A( ) = ∗ ( − δ−1) ∗ ( ) q 1 q n gn q n=0 and where, for all n ∈ N, the function gn is the never vanishing entire regular function whose restriction to the plane Ln = R + R[Im(δn)] is given by − zδ−1+ 1 z2δ−2···+ 1 zmn δ mn | ( ) = n 2 n mn n . gn Ln z e In particular, we can choose rn = n = mn = n for all n ∈ N. The above result is achieved in several steps. Some of them are inspired by, and similar to, the steps that lead to the complex version of the Weierstrass factorization theorem, while other are new and quite diverse, mainly due to the peculiarities of the quaternionic setting and the fact that the structure of the zero-set of the entire regular functions includes spherical zeros, which are not present in the case of the entire holomorphic functions. This article is organized as follows. After having presented some necessary preliminary results, in Sect.3, we study criteria to establish the convergence of infinite products of quaternions, and this latter necessity leads us to the (peculiar) definition of a quaternionic Logarithm. Then, in Sect.4, we investigate the uniform convergence of infinite products of regular functions. The results so found give us a tool to find, in Sect.5, conditions that can guarantee the uniform convergence of infinite products of regular functions to a regular function. In Sect. 6 we perform the study of different type of convergence-producing regular factors, some of 123 Ann Glob Anal Geom which arise only in the quaternionic setting. Section 7 is dedicated to give the definitions of spherical and isolated multiplicities for the zeros of a regular function; to do this, we follow the path which brought to the same definitions in the case of quaternionic regular polynomials.

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