Discrete Hardy Spaces for Bounded Domains in $${\mathbb {R}}^{n}$$ R
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Complex Analysis and Operator Theory
Discrete Hardy Spaces for Bounded Domains in Rn Paula Cerejeiras1 · Uwe Kähler1 · Anastasiia Legatiuk2 · Dmitrii Legatiuk2 Received: 18 April 2020 / Accepted: 13 October 2020 © The Author(s) 2020
Abstract Discrete function theory in higher-dimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as half-space, while the case of bounded domains generally remained unconsidered. Therefore, this paper presents the extension of the higherdimensional function theory to the case of arbitrary bounded domains in Rn . On this way, discrete Stokes’ formula, discrete Borel–Pompeiu formula, as well as discrete Hardy spaces for general bounded domains are constructed. Finally, several discrete Hilbert problems are considered. Keywords Discrete Dirac operator · Discrete monogenic functions · Discrete function theory · Discrete Cauchy transform · Discrete boundary value problems Mathematics Subject Classification 39A12 · 42A38 · 44A15
Communicated by Irene Sabadini. This article is part of the topical collection “Higher Dimensional Geometric Function Theory and Hypercomplex Analysis” edited by Irene Sabadini, Michael Shapiro and Daniele Struppa.
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Anastasiia Legatiuk [email protected] Paula Cerejeiras [email protected] Uwe Kähler [email protected] Dmitrii Legatiuk [email protected]
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CIDMA – Center for R & D in Mathematics and Applications, Universidade de Aveiro, Campus de Santiago, 3810-193 Aveiro, Portugal
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Chair of Applied Mathematics, Bauhaus-Universität Weimar, Coudraystr. 13B, 99423 Weimar, Germany 0123456789().: V,-vol
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P. Cerejeiras et al.
1 Introduction Construction of discrete analogues to the classical theory of monogenic functions has been an area of active research during last decades. The motivation for this construction has also evolved over the time: while initially the principal interest consisted in developing numerical methods based on integral representation formulae, see for example papers [9,11,12] and references therein, later, the growing importance of discrete modelling in various practical fields, e.g. [23], led to a genuine and native interest in discrete structures in the hypercomplex setting, see [2,7,10] and references therein. The main advantage of discrete modelling lies in the fact that certain properties of a continuous problem are exactly replicated on the discrete level and they are not just an approximation as in conventional numerical schemes, e.g. factorisation of the discrete Laplace operator by pair of discrete Cauchy–Riemann operators. Therefore, it is not surprising that studies of discrete function theory and other related theories in different settings have been presented by many authors. Thus, to keep the presentation short and being completely aware of such classical topic as theory of discrete analytic functions [20], we will focus only on works relevant to the paper and related to the hypercomplex
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