Cauchy Problems for Discrete Holomorphic Functions
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Complex Analysis and Operator Theory
Cauchy Problems for Discrete Holomorphic Functions Christer Oscar Kiselman1 Received: 17 April 2020 / Accepted: 12 August 2020 © The Author(s) 2020
Abstract We solve Cauchy problems for discrete holomorphic functions defined on the Gaussian integers, which leads to the existence of discrete holomorphic functions with arbitrarily fast growth. This proves that certain classes of functions are closed in the sense of mathematical morphology. Keywords Discrete holomorphic functions · Duality of convolution operators · Mathematical morphology
1 Introduction 1.1 Discrete Holomorphic Functions Functions on the ring of Gaussian integers Z[i] = Z + iZ ⊂ C that satisfy a discrete convolution equation corresponding to the Cauchy–Riemann equation have been studied since the 1940s. Rufus Philip Isaacs [6,7] developed a theory for these functions. He called them monodiffric, meaning that the quotients of differences in two different directions are equal. Those for the direction from 0 to 1 versus the direction from 0 to i (illustrated by the arrows ↑→) he called monodiffric of the first kind; those for the direction from 0 to 1 + i versus the orthogonal direction from 1 to i (illustrated by the arrows ) he called monodiffric of the second kind. He expressed the opinion that the latter “seemed less promising than the present course” [7: 258]. I therefore think that
Dedicated to the memory of Carlos Alberto Berenstein. Communicated by Daniel Aron Alpay. This article is part of the topical collection “In memory of Carlos A. Berenstein (1944–2019)” edited by Irene Sabadini and Daniele Struppa.
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Christer Oscar Kiselman [email protected]; [email protected] http://www.cb.uu.se/~kiselman Department of Information Technology, Uppsala University, P. O. Box 337, 751 05 Uppsala, Sweden 0123456789().: V,-vol
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it is justifiable to associate the monodiffric functions of the first kind with the name of Isaacs, as I did in my paper [9]. More recent research on this class includes that of Nakamura and Rosenfeld [17]. The monodiffric functions of the second kind were studied by Jacqueline Ferrand (1944) [5], which is the reason why I associated them with her name in my paper (2008) [10]. She called these functions préholomorphes. Later studies on this class include those by Duffin (1956) [4], Lovász (2000) [15], Kenyon (2000) [8] and Benjamini and Lovász (2000) [1]. For these two most studied classes, we shall prove that a Cauchy problem has a unique solution: for those of the first kind with data given on {z ∈ Z[i]; Im z = 0} ∪ {z ∈ Z[i]; Re z = 0 and Im z 0}; for those of the second kind with data given on the axes: Z ∪ iZ = {z ∈ Z[i]; Im z = 0} ∪ {z ∈ Z[i]; Re z = 0}. For the question discussed here the functions of the second kind are more symmetric than those of the first kind and require less work—you can compare Propositions 7.4 and 8.4. 1.2 Duality of Convolution Operators Duality is a term which represents a collection of ideas where two sets of mathema
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