On the Riemann Boundary Value Problem for Poly- and Meta-hyperanalytic Function Spaces over d-summable Curves
Abstract
Hyperanalytic functions, in the sense established by the mathematician Avron Douglis, are Douglis algebra-valued functions defined via a hypercomplex structure rather than the standard Cauchy-Riemann equations characteristic of traditional complex analysis. The classes of polyhyperanalytic and meta-hyperanalytic functions represent advanced generalizations of Douglis's analysis. They are employed in the study of partial differential equations and elasticity, extending the concept of the classical holomorphic function through higher-order iterations and non-homogeneous terms. The aim of this work is to find solvability conditions for a fundamental Riemann-type boundary value problem for spaces of poly-hyperanalytic and meta-hyperanalytic functions defined on an open, bounded, simply connected subset of the complex plane, where the boundary need only be a closed d-summable curve. In fractal geometry, d-summability is a geometric property used to define the boundaries of fractal domains, enabling advanced mathematical integration and calculus on complex structures defined by Jenny Harrison and Alec Norton.
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BibTeX
@misc{dai2026riemann,
title = {{On the Riemann Boundary Value Problem for Poly- and Meta-hyperanalytic Function Spaces over d-summable Curves}},
author = {Yan Dai and Juan Bory-Reyes and Fuli He},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30634},
eprint = {2608.30634},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30634},
}
RIS
TY - UNPB
TI - On the Riemann Boundary Value Problem for Poly- and Meta-hyperanalytic Function Spaces over d-summable Curves
AU - Yan Dai
AU - Juan Bory-Reyes
AU - Fuli He
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30634
UR - https://arxiv.org/abs/2608.30634
AB - Hyperanalytic functions, in the sense established by the mathematician Avron Douglis, are Douglis algebra-valued functions defined via a hypercomplex structure rather than the standard Cauchy-Riemann equations characteristic of traditional complex analysis. The classes of polyhyperanalytic and meta-hyperanalytic functions represent advanced generalizations of Douglis's analysis. They are employed in the study of partial differential equations and elasticity, extending the concept of the classical holomorphic function through higher-order iterations and non-homogeneous terms. The aim of this work is to find solvability conditions for a fundamental Riemann-type boundary value problem for spaces of poly-hyperanalytic and meta-hyperanalytic functions defined on an open, bounded, simply connected subset of the complex plane, where the boundary need only be a closed d-summable curve. In fractal geometry, d-summability is a geometric property used to define the boundaries of fractal domains, enabling advanced mathematical integration and calculus on complex structures defined by Jenny Harrison and Alec Norton.
ER -