# Octonionic Dirac operators in bounded lattices: Convergence and Plemelj
projections

Guangbin Ren<EMAIL_ADDRESS>School of Mathematical Sciences, University of
Science and Technology of China, Hefei 230026, China  and  Xin Zhao
<EMAIL_ADDRESS>Department of Mathematics, University of Science and
Technology of China, Hefei 230026, China

###### Abstract.

This article explores octonionic analysis on the lattice $\mathbb{Z}^{8}$,
emphasizing the octonionic discrete Cauchy integral within a bounded domain,
the Sokhotski-Plemelj jump formulas, and the convergence of discrete regular
functions. We address the challenge of articulating the discrete Cauchy-
Pompeiu integral formula, particularly the complexities tied to the associator
of octonions. Adopting an innovative approach, we assimilate complicating
terms into integral kernels, leading to distinct surface and volume kernels. A
new ’star product’ arises in handling octonionic function multiplications. Our
study connects closely with the Teodorescu operator and singular kernel
management, utilizing a Fourier variant with 24 unique singular points.
Drawing from the foundational work of Shivakumar and Wong about the asymptotic
expansion of the Fourier transform, we bridge the relationship between the
Cauchy and Teodorescu integrals. This allows us to provide quantitative
estimates for these kernels, which prove to be pivotal in the theory of the
discrete regular extensions. The research culminates in the revelation that a
continuous octonionic function is regular precisely when it is a scale limit
of discrete regular functions.

###### Key words and phrases:

Discrete complex analysis, octonions, Plemej projection, Dirac operators,
Cauchy integral, boundary value problem

###### 2010 Mathematics Subject Classification:

Primary: 30G35; 39A12; Secondary: 17D99; 42A38; 47G10; 65E05

This work was supported by the NNSF of China (12171448).

###### Contents

  1. 1 Introduction
  2. 2 Foundations of Discrete Octonionic Analysis
  3. 3 Octonionic Analysis: The Discrete Stokes Theorem
  4. 4 The Cauchy-Pompeiu Formula and Its Inverse on $\mathbb{Z}^{8}_{h}$
  5. 5 Estimation of the Discrete Cauchy Kernel
  6. 6 Sokhotski-Plemelj formula
  7. 7 Convergence of Discrete Regular Octonionic Functions

##  1\. Introduction

In recent years, with the advancement of physics and computer science, there
has been an increasing reliance on non-associative mathematical theories and
discrete mathematics. For instance, the $M$-theory of cosmological models
requires the octonion theory [36], the Ising model in statistical physics
necessitates discrete complex analysis [34], and computer graphics hinge on
discrete differential geometry [16]. Both non-associative and discrete
mathematics have witnessed remarkable advancements, such as in operator theory
[1], octonion functional analysis [19], and $G_{2}$ manifold theory [21].
However, the road ahead is riddled with challenges, from the uncharted
territories of matrix implementations in non-associative algebra [5, 6] to the
seemingly vast divide between discrete and continuous mathematics [27].

Early forays into this domain can be traced back to the seminal works of
Isaacs, Ferrand, and Duffin on complex analysis on planar graphs [20, 14, 13].
For further development, see also [3, 26, 4, 15, 22].

This paper’s ambition is to seamlessly integrate non-associative and discrete
mathematics. Our focus is sharpened on the analytical theory of discrete
octonionic Dirac operators in bounded lattices. We venture deep into two
pivotal questions:

Discrete Octonionic Function Boundary Theory: How can we aptly mold the
discrete Cauchy integral and the discrete Plemelj projection operators to
reflect the essence of their continuous analogs?

Convergence in the Scaling Limit: Is there a smooth progression of a sequence
of discrete octonionic regular functions towards their continuous versions as
lattice granularity diminishes, and vice versa?

######  Remark 1.

The landscape of octonions, marked by its non-associative nature, has yet to
see a definitive blueprint for the discrete Cauchy integral and the discrete
Plemelj projection operators. Krausshar and Legatiuk [24] took pioneering
steps in this direction, a journey also echoed in [23].

The theme of convergence in discrete harmonic functions was notably charted by
Courant, Friedrichs, and Lewy for square lattices [11], by Chelkak-Smirnov for
rhombic lattices [9], and Skopenkov for orthogonal quadrilateral lattices
[33].

Drawing from the application of discrete holomorphic functions in the Ising
models, Smirnov posited several open problems, with a keen spotlight on the
convergence quandary [35]. In [30], the authors explored the convergence
within the quaternionic domain for square lattices [30]. In this study, we
venture into this topic in the non-associative setting, specifically examining
octonions within bounded square lattices. This tackles a question raised by
Skopenkov back in 2013 [33, Problem 5.9(3)(6)].

Navigating the terrain of octonions brings its own set of intricacies:

  * •

The onus of constructing an analytical theory, mirroring its continuous
counterpart in bounded lattices, is substantial. This is particularly true
when venturing into the domain of boundary theories, where concepts such as
domain boundaries, boundary normal vectors, non-tangential limits, and
boundary measures in the discrete realm become complex to transpose.

  * •

The analytical scaffolding of the Dirac operator rests on deciphering its
foundational solution–a cornerstone for diving into singular integral
theories. The disparity between complex and octonion analysis, especially at
their foundational crossroads, adds layers of complexity. The Fourier
interpretations for each present different challenges, making evaluations for
multi-singular scenarios intricate.

  * •

Translating the continuous Stokes theorem into its non-associative avatar
introduces convoluted terms that are cumbersome to handle [25]. This
complexity is magnified in the discrete iteration, posing significant hurdles.

  * •

The realm of bounded lattices is replete with its own challenges, distinct
from the upper half lattices [7]. The intricacies emerge mainly due to
boundary irregularities and mesh size dependencies.

Our roadmap draws inspiration from the discrete quaternion analysis. The birth
of discrete quaternionic analysis on equidistant grids can be attributed to
Gürlebeck and Sprössig [18]. The field has since witnessed contributions by
Cerejeiras–Kähler–Ku–Sommen [8]. Our methodology aligns with the paradigm set
forth in [29]. Although formulas based on summation might seem straightforward
for discrete analysis, reframing them through the lens of integration provides
a more transparent connection between discrete and continuous realms. This
reformulation facilitates the emergence of parallel notions like the ’discrete
boundary measure’ and ’discrete normal vector’, which promote a more
harmonious integration of discrete and continuous methodologies.

Building the discrete Cauchy-Pompeiu integral formula presents significant
challenges, especially when avoiding intricate details associated with the
octonionic associator. In our methodology, we unveil the ’star product’
technique, a pivotal mechanism that forms a link between the two integral
kernels. Based on this foundation, we elucidate the discrete integral by
referencing the discrete surface kernel. At this juncture, traditional
multiplication transitions to the star product, guiding us in our
investigation of the Teodorescu operator, which is further enriched by the
core solution of the discrete Dirac operator. Consequently, the formulation of
the Cauchy-Pompeiu integral can be presented as

$\displaystyle\chi_{B}(y)\bm{f}(y)=\int_{\partial
B}\bm{K}^{h}(x,y)*\bm{f}(x)dS(x)+\int_{B}\bm{E}^{h}(y-x)(\bm{D}^{h}\bm{f})(x)dV^{h}(x),$

for any function $\bm{f}:\overline{B}\longrightarrow\mathbb{O}$ with $B$ being
a bounded set in $\mathbb{Z}^{8}_{h}$; see Theorem 3.

The bedrock of our research is anchored in the insights from Shivakumar and
Wong’s exploration of the asymptotic expansions of Fourier transforms [32,
38]. This provides the relations between the discrete and continuous
fundamental solutions of octonionic Dirac operators:

$|\bm{E}^{h}(x)-\omega(h^{-1}x)\bm{E}(x)|\leqslant\dfrac{Ch}{|x|^{8}+h^{8}}.$

for any $h>0$ and $x\in\mathbb{Z}^{8}_{h}$; see Theorem 9. The traditional
approach by Thomée [37] is not applicable to the octonionic context, as it is
tailored for scenarios with a single singular point, whereas the octonionic
setting presents the complexity of handling 24 distinct singular points.

In subsequent sections, akin to its continuous peer, we embark on crafting the
orthogonal projection within the Plemelj boundary jump paradigm:

$\bm{C}^{h}_{\partial
B}\bm{f}(y)=\frac{1}{2}\left(\pm\bm{f}(y)+\bm{S}^{h}_{\partial
B}\bm{f}(y)\right),\quad\forall\ y\in\partial^{\pm}B,$

for any function $f\in M(\partial B,\mathbb{O})$ with $B$ being a bounded
lattice in $\mathbb{Z}^{8}_{h}$; see Theorem 10.

In our final discussion, drawing from the aforementioned Cachy-Pompeiu
integral formula, we introduce a compelling theory: a continuous octonionic
function, when viewed as a scaling limit of discrete regular functions,
inherently exhibits regularity.

##  2\. Foundations of Discrete Octonionic Analysis

In this section, we lay down the foundational concepts and notations pertinent
to discrete octonionic analysis.

###  2.1. Octonion Algebra

The octonion algebra, denoted as $\mathbb{O}$, stands as the largest
composition algebra, as confirmed by Hurwitz’s Theorem [2]. Representing an
eight-dimensional $\mathbb{R}$-space, it is constructed on the standard basis:

$\bm{e}_{0}=1,\bm{e}_{1},\cdots,\bm{e}_{7}$

and adheres to the relations:

$\bm{e}_{i}\bm{e}_{j}=\begin{cases}\bm{e}_{j},\quad&i=0,\\\
\bm{e}_{i},\quad&j=0,\\\
-\delta_{i,j}+\varepsilon_{ijk}\bm{e}_{k},&otherwise,\end{cases}$

where $\delta_{i,j}$ symbolizes the Kronecker delta, and $\varepsilon_{ijk}$
is an entirely antisymmetric tensor that equals 1 when the sequence $ijk$ is
one of:

$123,145,176,246,257,347,365.$

The octonions, by nature, are non-commutative and non-associative, as
evidenced by the relation:

$(\bm{e}_{1}\bm{e}_{2})\bm{e}_{4}=-\bm{e}_{1}(\bm{e}_{2}\bm{e}_{4}).$

Further, the relationship can be generalized as:

(2.1)
$(\bm{e}_{i}\bm{e}_{j})\bm{e}_{k}=\begin{cases}\bm{e}_{i}(\bm{e}_{j}\bm{e}_{k}),&k=0,i,j,i\oplus
j,\\\ \\\ -\bm{e}_{i}(\bm{e}_{j}\bm{e}_{k}),&otherwise.\end{cases}$

Here $i\oplus j$ is the standard basis in $\mathbb{Z}_{8}$; see [28].

######  Definition 1.

The associator for any elements $\bm{a,b,c}\in\mathbb{O}$ is defined as

$\bm{[a,b,c]:=(ab)c-a(bc).}$

######  Lemma 1 ([31, 2]).

The octonion algebra $\mathbb{O}$ exhibits both alternativity and flexibility,
implying that

$\bm{[a,a,b]=[b,a,a]}=0$

and

$\bm{[a,b,a]}=0$

for any elements $\bm{a,b}\in\mathbb{O}$.

Additionally, the octonions satisfy the Moufang identities:

######  Lemma 2.

For any $\bm{a,b,c}\in\mathbb{O},$ one has

$\bm{(aba)c=a(b(ac)),\quad c(aba)=((ca)b)a,\quad a(bc)a=(ab)(ca).}$

###  2.2. Basic concepts in the discrete octonionic analysis

Our analysis centers on a discrete function theory mapped onto the eight-
dimensional grids $\mathbb{Z}^{8}_{h}=(h\mathbb{Z})^{8},h>0$, which serves as
a discretization of $\mathbb{R}^{8}$. This grid is embedded in the octonion
space $\mathbb{O}$ using:

$\bm{x}=\sum_{i=0}^{7}\bm{e}_{i}x_{i}$

for any point $x=(x_{0},\cdots,x_{7})\in\mathbb{Z}^{8}_{h}$. Here,
$\\{\bm{e}_{0},\cdots,\bm{e}_{7}\\}$ signifies the standard basis of
$\mathbb{O}$ over the $\mathbb{R}$-space.

######  Definition 2.

Given a subset $B$ of $\mathbb{Z}^{8}_{h}$, $M(B,\mathbb{O})$ represents the
collection of all mappings from $B$ to $\mathbb{O}$. The discrete Cauchy-
Fueter operator, denoted as $\bm{D}^{h}$, acts as a transformation from
$M(B,\mathbb{O})$ to $M(B,\mathbb{O})$ and is expressed as:

$\bm{D}^{h}=\sum_{l=0}^{7}\bm{e}_{i}\partial^{h}_{l},$

with

$\partial^{h}_{l}=\frac{1}{2}(\partial_{l}^{+,h}+\partial_{l}^{-,h}).$

Here, the forward and backward difference operators, $\partial_{l}^{+,h}$ and
$\partial_{l}^{-,h}$, in relation to the $i$-th variable $x_{i}$, are defined
respectively as:

$\displaystyle\partial_{l}^{+,h}\bm{f}(x)$
$\displaystyle=\frac{1}{h}[\bm{f}(x+he_{l})-\bm{f}(x)],$
$\displaystyle\partial_{l}^{-,h}\bm{f}(x)$
$\displaystyle=\frac{1}{h}[\bm{f}(x)-\bm{f}(x-he_{l})]$

for any function $\bm{f}$ in $M(B,\mathbb{O})$.

We also consider different types of discrete boundaries, opting for a two-
layered boundary in this context.

######  Definition 3.

For a subset $B$ within $\mathbb{Z}^{8}_{h}$, the boundary $\partial B$ of $B$
is defined as:

$\partial B=\\{x\in\mathbb{Z}^{8}_{h}|N(x)\cap
B\neq\emptyset,N(x)\cap(\mathbb{Z}^{8}_{h}\setminus B)\neq\emptyset\\}$

where $N(x)$ denotes the neighborhood of $x$:

$N(x)=\\{x,x\pm he_{0},\cdots,x\pm he_{7}\\}.$

We can also define the discrete closure $\overline{B}$ and the interior
$B^{\circ}$ of $B$ as:

$\overline{B}=B\cup\partial B,\quad B^{\circ}=B\setminus\partial B.$

In this study, we represent the set of functions
$\bm{f}:\overline{B}\rightarrow\mathbb{O}$ by $M(\overline{B},\mathbb{O})$.

######  Definition 4.

A discrete function $\bm{f}$ in $M(\overline{B},\mathbb{O})$ is termed
discrete octonionic regular if $\bm{D}^{h}\bm{f}=0$.

Additionally, we delve into the concepts of discrete boundary measure and
discrete outward normal vector in relation to $\mathbb{Z}^{8}_{h}$.

######  Definition 5.

For a subset $B$ in $\mathbb{Z}^{8}_{h}$, the discrete outward normal vector
at a boundary point $x$ of $\partial B$ is given by:

$n=(n_{0}^{+},n_{0}^{-},\cdots,n_{7}^{+},n_{7}^{-}),$

where:

$n_{l}^{\pm}(x)=\frac{-2\partial_{l}^{\pm,h}\chi_{B}(x)}{\sqrt{\sum_{l=0}^{7}[\left(\partial_{l}^{+,h}\chi_{B}\right)^{2}+\left(\partial_{l}^{-,h}\chi_{B}\right)^{2}]}}.$

The discrete octonionic outside normal vector $\bm{n}$ on $\partial B$ is then
defined as:

$\bm{n}=\sum_{l=0}^{7}\frac{1}{2}(n_{l}^{+}+n_{l}^{-})\bm{e}_{l}.$

######  Remark 2.

Both $s(x)$ and $n_{l}^{\pm}$ are functions defined on the boundary $\partial
B\subset\mathbb{Z}^{8}_{h}$. By extending these functions to the entire space
$\mathbb{Z}^{8}_{h}$ through zero-extension, we can interpret Lemma 5 in a
point-wise manner. This provides ample convenience in proving theorems
throughout this article. This approach stands in contrast to the methodology
used in [29] for discrete quaternion analysis, wherein distributions were
employed.

##  3\. Octonionic Analysis: The Discrete Stokes Theorem

The realm of octonionic analysis presents a unique version of the continuous
Stokes theorem, which differs notably from the classical interpretation found
in complex analysis. This octonionic variant introduces an added term linked
to the associator, a concept detailed in [25].

Various methodologies have been explored concerning the discrete integral
theory across different contexts. For insights into the complex and
quaternionic scenarios, we refer to references [9], [10], [17], and [29].

###  3.1. Multiplicative Discrete Derivative Formula

A significant challenge in the discrete Stokes theorem is the complex nature
of the discrete derivative formula under multiplication, especially when
contrasted against the continuous scenario. To navigate this complexity, we
introduce specific shift operators to assimilate the extra terms.

######  Definition 6.

Given any point $y$ in $\mathbb{Z}^{8}_{h}$, the translation operator, denoted
by $\tau_{y}:M(B,\mathbb{O})\longrightarrow M(B,\mathbb{O})$, is defined as

$\tau_{y}\bm{f}(x):=\bm{f}(x-y)$

for any $\bm{f}\in M(B,\mathbb{O}).$

######  Lemma 3 (Leibniz rule I).

Let $B$ be a subset in $\mathbb{Z}^{8}_{h}.$ Then for any $\bm{f},\bm{g}\in
M(B,\mathbb{O}),$ we have

(3.1) $\displaystyle\partial^{+,h}_{l}(\bm{f}\bm{g})$
$\displaystyle=\partial_{l}^{+,h}\bm{f}\bm{g}+\tau_{-he_{l}}\bm{f}\partial_{l}^{+,h}\bm{g},$
$\displaystyle\partial^{+,h}_{l}(\bm{f}\bm{g})$
$\displaystyle=\partial^{+,h}_{l}\bm{f}\tau_{-he_{l}}\bm{g}+\bm{f}\partial_{l}^{+,h}\bm{g},$
$\displaystyle\partial^{-,h}_{l}(\bm{f}\bm{g})$
$\displaystyle=\partial_{l}^{-,h}\bm{f}\bm{g}+\tau_{he_{l}}\bm{f}\partial_{l}^{-,h}\bm{g},$
$\displaystyle\partial^{-,h}_{l}(\bm{f}\bm{g})$
$\displaystyle=\partial_{l}^{-,h}\bm{f}\tau_{he_{l}}\bm{g}+\bm{f}\partial_{i}^{-,h}\bm{g}.$

###### Proof.

For the first two equalities, by definition we have

$\displaystyle\partial_{l}^{+,h}(\bm{f}\bm{g})(x)$
$\displaystyle=\frac{1}{h}(\bm{f}(x+he_{l})\bm{g}(x+he_{l})-\bm{f}(x)\bm{g}(x))$
$\displaystyle=\frac{1}{h}(\bm{f}(x+he_{l})\bm{g}(x+he_{l})-\bm{f}(x+he_{l})\bm{g}(x)+\bm{f}(x+he_{l})\bm{g}(x)-\bm{f}(x)\bm{g}(x))$
$\displaystyle=\tau_{-he_{l}}\bm{f}(x)\partial^{+,h}_{l}\bm{g}(x)+\partial_{l}^{+,h}\bm{f}(x)\bm{g}(x),$

and

$\displaystyle\partial_{l}^{+,h}(\bm{f}\bm{g})(x)$
$\displaystyle=\frac{1}{h}(\bm{f}(x+he_{l})\bm{g}(x+he_{l})-\bm{f}(x)\bm{g}(x))$
$\displaystyle=\frac{1}{h}(\bm{f}(x+he_{l})\bm{g}(x+he_{l})-\bm{f}(x)\bm{g}(x+he_{l})+\bm{f}(x)\bm{g}(x+he_{l})-\bm{f}(x)\bm{g}(x))$
$\displaystyle=\partial_{l}^{+,h}\bm{f}(x)\bm{g}(x+he_{l})+\bm{f}(x)\partial_{l}^{+,h}\bm{g}(x).$

The proofs for the remaining two equalities follow a similar approach. ∎

To unify the variables of functions on the right side of (3.1), we have the
following alternatives.

######  Lemma 4 (Leibniz rule II).

(3.2) $\displaystyle\partial^{+,h}_{l}(\bm{f}\bm{g})$
$\displaystyle=\partial^{+,h}_{l}\bm{f}\tau_{-he_{l}}\bm{g}+\bm{f}\partial_{l}^{-,h}\tau_{-he_{l}}\bm{g},$
$\displaystyle\partial^{+,h}_{l}(\bm{f}\bm{g})$
$\displaystyle=\partial_{l}^{-,h}\tau_{-he_{l}}\bm{f}\bm{g}+\tau_{-he_{l}}\bm{f}\partial_{l}^{+,h}\bm{g},$
$\displaystyle\partial^{-,h}_{l}(\bm{f}\bm{g})$
$\displaystyle=\partial_{l}^{-,h}\bm{f}\tau_{he_{l}}\bm{g}+\bm{f}\partial_{i}^{+,h}\tau_{he_{l}}\bm{g},$
$\displaystyle\partial^{-,h}_{l}(\bm{f}\bm{g})$
$\displaystyle=\partial_{l}^{+,h}\tau_{he_{l}}\bm{f}\bm{g}+\tau_{he_{l}}\bm{f}\partial_{l}^{-,h}\bm{g}.$

###### Proof.

This follows from the fact that

(3.3) $\displaystyle\partial_{l}^{+,h}$
$\displaystyle=\partial_{l}^{-,h}\tau_{-he_{l}},$
$\displaystyle\partial_{l}^{-,h}$
$\displaystyle=\partial_{l}^{+,h}\tau_{he_{l}}.$

∎

###  3.2. Discrete Stokes theorem

In this subsection, we will employ the discrete derivative formula in the
context of multiplication to formulate a discrete Stokes theorem without the
supplementary term. This is in alignment with the methodology delineated in
[29, 30]. As highlighted in [29, 30], for a bounded domain
$B\in\mathbb{R}^{8}$ with a $C^{1}$-boundary, a unique solution exists for the
Gauss system represented as:

(3.4) $\begin{cases}n_{l}dS=-\frac{\partial}{\partial x_{l}}\chi_{B}dV,\\\
\sum\limits_{l=0}^{7}n_{l}^{2}=\chi_{B}.\end{cases}$

Here, $dS$ denotes a non-negative regular Borel measure, defined by the
restriction of the Lebesgue measure on $\partial B$.

Subsequently, we derive a discrete counterpart of this theorem. Throughout
this paper, $dV^{h}$ symbolizes the Haar measure on the group
$\mathbb{Z}^{8}_{h}$. More specifically, for any function
$\bm{f}:\mathbb{Z}^{8}_{h}\longrightarrow\mathbb{O}$, we define

$\int_{\mathbb{Z}^{8}_{h}}\bm{f}dV^{h}=\sum\limits_{x\in\mathbb{Z}^{8}_{h}}\bm{f}(x)h^{8}.$

######  Definition 7.

Given a set $B$ in $\mathbb{Z}^{8}_{h}$, the discrete boundary measure,
$S_{\partial B}$, on $\partial B$ is defined by

$S_{\partial B}(U):=\sum\limits_{x\in U}s(x)$

for any $U\subset\partial B$ for any subset $U$ of $\partial B$. Here, the
function $s:\partial B\longrightarrow\mathbb{O}$ is defined as

$s(x):=\frac{h^{8}}{2}\sqrt{\sum\limits_{l=0}^{7}\left((\partial_{l}^{+,h}\chi_{B})^{2}+(\partial_{l}^{-,h}\chi_{B})^{2}\right)}.$

######  Lemma 5.

Let $B$ be a set in $\mathbb{Z}^{8}_{h}$. Let
$n=(n_{0}^{+},n_{0}^{-},\cdots,n_{7}^{+},n_{7}^{-})$ be the external normal
vector on $\partial B$ and let $S=S_{\partial B}$ represent the boundary
measure of $\partial B$. Then, we have

(3.5) $\begin{cases}n_{l}^{\pm}dS=-\partial_{l}^{\pm,h}\chi_{B}dV^{h},\\\ \\\
\sum\limits_{l=0}^{7}\left((n_{l}^{+})^{2}+(n_{l}^{-})^{2}\right)=4\chi_{B}.\end{cases}$

###### Proof.

We commence by proving the first equation in (3.5).

Suppose $x\notin\partial B$. Then, the relation $x\in B$ is equivalent to
$x\pm he_{l}\in B$. This yields

$\partial_{l}^{\pm,h}\chi_{B}(x)=0.$

Given the zero-extension, this implies:

$dS(x)=0$

Consequently, we establish:

$n_{l}^{\pm}dS=-\partial_{l}^{\pm,h}\chi_{B}dV^{h}\quad\text{on}\quad\mathbb{Z}^{8}\setminus\partial
B.$

For every $x\in\partial B$, the following holds:

$\displaystyle n_{l}^{\pm}(x)dS(x)$ $\displaystyle=n_{l}^{\pm}(x)s(x)$
$\displaystyle=\frac{-2\partial_{l}^{\pm,h}\chi_{B}(x)}{\sqrt{\sum_{i=0}^{7}\left((\partial_{l}^{+,h}\chi_{B})^{2}+(\partial_{l}^{-,h}\chi_{B})^{2}\right)}}\cdot\frac{h^{8}}{2}\sqrt{\sum_{i=0}^{7}\left((\partial_{l}^{+,h}\chi_{B})^{2}+(\partial_{l}^{-,h}\chi_{B})^{2}\right)}$
$\displaystyle=-2\partial_{l}^{\pm,h}\chi_{B}(x)h^{8}$
$\displaystyle=-2\partial_{l}^{\pm,h}\chi_{B}(x)dV^{h}.$

To ascertain the second equation in (3.5), we employ the predefined notion
that

$\sum_{i=0}^{7}\left((n_{l}^{+})^{2}+(n_{l}^{-})^{2}\right)\bigg{|}_{x\in\partial
B}=4.$

Given that $n_{l}^{\pm}$ is null outside $\mathbb{Z}^{8}_{h}\setminus\partial
B$, it follows:

$\sum_{i=0}^{7}\left((n_{l}^{+})^{2}+(n_{l}^{-})^{2}\right)\bigg{|}_{x\notin\partial
B}=0.$

This concludes our proof. ∎

######  Theorem 1 (Divergence Principle).

Let $B$ be a bounded set in $\mathbb{Z}^{8}_{h}$. Then for every
$l=0,\cdots,7$, we have

(3.6) $\int_{\partial
B}\bm{f}n_{l}^{\pm}dS=\int_{B}\partial^{\mp,h}_{l}\bm{f}dV^{h}$

for any function $\bm{f}\in M(B,\mathbb{O})$.

###### Proof.

Extending the function $\bm{f}$ to encompass the entire space
$\mathbb{Z}^{8}_{h}$, the relationship (3.5) yields

(3.7) $\int_{\partial
B}\bm{f}n_{l}^{\pm}dS=\int_{\mathbb{Z}^{8}_{h}}\bm{f}n_{l}^{\pm}dS_{\partial
B}=-\int_{\mathbb{Z}^{8}_{h}}\bm{f}\partial_{l}^{\pm,h}\chi_{B}dV^{h}.$

From the relation

(3.8)
$\displaystyle-\int_{\mathbb{Z}^{8}_{h}}\bm{f}\partial_{l}^{+,h}\chi_{B}dV^{h}$
$\displaystyle=-\frac{1}{h}\int_{\mathbb{Z}^{8}_{h}}\bm{f}(x)(\chi_{B}(x+he_{l})-\chi_{B}(x))dV^{h}$
$\displaystyle=\int_{\mathbb{Z}^{8}_{h}}\partial_{l}^{-,h}\bm{f}(x)\chi_{B}(x)dV^{h}(x)$
$\displaystyle=\int_{B}\partial_{l}^{-,h}\bm{f}dV^{h}$

and a parallel derivation indicating:

(3.9)
$-\int_{\mathbb{Z}^{8}_{h}}\bm{f}\partial_{l}^{-,h}\chi_{B}dV^{h}=\int_{B}\partial_{l}^{+,h}\bm{f}dV^{h},$

integrating the results of (3.7), (3.8), and (3.9) completes the proof. ∎

######  Corollary 1.

For a bounded set $B$ in $\mathbb{Z}^{8}_{h}$, the following relationships
hold:

(3.10) $\displaystyle\int_{\partial B}\bm{f}\bm{n}dS$
$\displaystyle=\int_{B}\bm{f}\bm{D}^{h}dV^{h},\ \int_{\partial
B}\bm{n}\bm{f}dS$ $\displaystyle=\int_{B}\bm{D}^{h}\bm{f}dV^{h}$

for any function $\bm{f}\in M(B,\mathbb{O})$.

###### Proof.

This follows directly from Theorem 1. ∎

######  Theorem 2 (Stokes Formula in Discrete Octonionic setting).

For a bounded set $B$ in $\mathbb{Z}^{8}_{h}$, for each $l=0,\cdots,7$ and for
any functions $\bm{f},\bm{g}\in M(B,\mathbb{O})$, the following relations are
true

(3.11) $\displaystyle\int_{\partial B}\bm{f}n_{l}^{+}\bm{g}dS$
$\displaystyle=\int_{B}\partial_{l}^{-,h}\bm{f}\bm{g}+\tau_{he_{l}}\bm{f}\partial_{l}^{-,h}\bm{g}dV^{h},$
$\displaystyle\int_{\partial B}\bm{f}n_{l}^{+}\bm{g}dS$
$\displaystyle=\int_{B}\partial_{l}^{-,h}\bm{f}\tau_{he_{l}}\bm{g}+\bm{f}\partial_{i}^{-,h}\bm{g}dV^{h},$
$\displaystyle\int_{\partial B}\bm{f}n_{l}^{-}\bm{g}dS$
$\displaystyle=\int_{B}\partial_{l}^{-,h}\bm{f}\bm{g}+\tau_{he_{l}}\bm{f}\partial_{l}^{-,h}\bm{g}dV^{h},$
$\displaystyle\int_{\partial B}\bm{f}n_{l}^{-}\bm{g}dS$
$\displaystyle=\int_{B}\partial^{+,h}l\bm{f}\tau{-he_{l}}\bm{g}+\bm{f}\partial_{l}^{+,h}\bm{g}dV^{h}.$

###### Proof.

By Theorem 1, we have

(3.12) $\int_{\partial
B}\bm{f}n_{l}^{+}\bm{g}dS=\int_{B}\partial^{-,h}l(\bm{f}\bm{g})dV^{h}.$

From relation (3.1), it is evident that:

(3.13) $\displaystyle\partial_{l}^{-,h}(\bm{f}\bm{g})$
$\displaystyle=\partial_{l}^{-}\bm{f}\bm{g}+\tau{he_{l}}\bm{f}\partial_{i,h}^{-}\bm{g},$
$\displaystyle\partial_{l}^{-,h}(\bm{f}\bm{g})$
$\displaystyle=\partial_{l}^{-,h}\bm{f}\tau_{he_{l}}\bm{g}+\bm{f}\partial_{i}^{-,h}\bm{g}.$

Incorporating results from (3.12) and (3.13), we derive:

$\displaystyle\int_{\partial B}\bm{f}n_{l}^{+}\bm{g}dS$
$\displaystyle=\int_{B}\partial_{l}^{-,h}\bm{f}\bm{g}+\tau_{he_{l}}\bm{f}\partial_{l}^{-,h}\bm{g}dV^{h},$
$\displaystyle\int_{\partial B}\bm{f}n_{l}^{+}\bm{g}dS$
$\displaystyle=\int_{B}\partial_{l}^{-,h}\bm{f}\tau_{he_{l}}\bm{g}+\bm{f}\partial_{i}^{-,h}\bm{g}dV^{h}.$

The proof of the remaining relationships in (3.11) follows a similar
methodology. ∎

##  4\. The Cauchy-Pompeiu Formula and Its Inverse on $\mathbb{Z}^{8}_{h}$

In this section, our focus lies on understanding the Cauchy-Pompeiu formula
and its inverse in the realm of discrete octonionic analysis.

For this purpose, we utilize the following relation:

$(\bm{e}_{i}\bm{e}_{j})\bm{e}_{k}=\begin{cases}\bm{e}_{i}(\bm{e}_{j}\bm{e}_{k}),&\text{when
}k=0,i,j,i\oplus j,\\\ \\\
-\bm{e}_{i}(\bm{e}_{j}\bm{e}_{k}),&\text{otherwise}.\end{cases}$

This equation stems from (2.1). Note that $\oplus$ represents the standard
addition in $\mathbb{Z}_{8}$. It is noteworthy that, in general,

$(\bm{e}_{i}\bm{e}_{j})\bm{e}_{k}\neq-\bm{e}_{i}(\bm{e}_{j}\bm{e}_{k}).$

######  Definition 8.

The discrete Laplacian
$\Delta^{h}:M(\mathbb{Z}^{8}_{h},\mathbb{O})\longrightarrow
M(\mathbb{Z}^{8}_{h},\mathbb{O})$ is defined by

$\Delta^{h}=\sum\limits_{i=0}^{7}\partial_{l}^{h}\partial_{l}^{h}.$

For $h=1$, we have

$\Delta^{1}\bm{f}(x)=\frac{1}{4}\sum\limits_{i=0}^{7}(\bm{f}(x+2e_{i})+\bm{f}(x-2e_{i})-2\bm{f}(x))$

for any $\bm{f}\in M(\mathbb{Z}^{8}_{h},\mathbb{O}).$ The fundamental solution
of $\Delta^{1}$ is given by

(4.1)
$F^{1}(x)=-\frac{1}{(2\pi)^{8}}\int_{[-\pi,\pi]^{8}}{\left(\sum\limits_{l=0}^{7}\sin^{2}u_{l}\right)^{-1}}e^{i\sum\limits_{l=0}^{7}u_{l}x_{l}}du,$

such that

(4.2) $\Delta^{1}F^{1}(x)=\delta^{1}_{0}(x)$

where $\delta_{0}^{1}(x)$ is the discrete Dirac delta function defined by

$\delta_{0}^{1}(x)=\begin{cases}1,\quad x=0,\\\ \\\ 0,\quad x\neq
0.\end{cases}$

Indeed, it can be shown that

$\Delta^{1}\left(e^{i\sum\limits_{l=0}^{7}u_{l}x_{l}}\right)=\left(-\sum\limits_{l=0}^{7}\sin^{2}u_{l}\right)e^{i\sum\limits_{l=0}^{7}u_{l}x_{l}},$

which leads to

(4.3) $\displaystyle\Delta^{1}F^{1}(x)$
$\displaystyle=-\frac{1}{(2\pi)^{8}}\int_{[-\pi,\pi]^{8}}\left(\sum\limits_{l=0}^{7}\sin^{2}u_{l}\right)^{-1}\left(-\sum\limits_{l=0}^{7}\sin^{2}u_{l}\right)e^{i\sum\limits_{l=0}^{7}u_{l}x_{l}}du$
$\displaystyle=\frac{1}{(2\pi)^{8}}\int_{[-\pi,\pi]^{8}}e^{i\sum\limits_{l=0}^{7}u_{l}x_{l}}du$
$\displaystyle=\delta_{0}^{1}(x).$

######  Definition 9.

A function $\bm{E}^{h}\in M(\mathbb{Z}^{8}_{h},\mathbb{O})$ is called the
fundamental solution of $\bm{D}^{h}$ if

(4.4) $\bm{D}^{h}\bm{E}^{h}=\delta_{0}^{h}(x)$

in a point-wise sense, where $\delta_{0}^{h}(x)$ is the discrete Dirac delta
function defined by

$\delta_{0}^{h}(x)=\begin{cases}\frac{1}{h^{8}},\quad x=0,\\\ \\\ 0,\quad
x\neq 0\end{cases}.$

Notably, (4.4) holds in a point-wise sense, leading to

(4.5) $\delta^{h}_{0}(x)=\frac{1}{h^{8}}\chi_{\\{0\\}}(x).$

This contrasts with the continuous case.

######  Lemma 6.

The fundamental solution $\bm{E}^{h}(x)$ of $\bm{D}^{h}$ satisfies

$\bm{E}^{h}(x)=\frac{1}{h^{7}}\bm{E}^{1}(\frac{x}{h}),$

where

(4.6)
$\bm{E}^{1}(x):=-\frac{1}{(2\pi)^{8}}\int_{[-\pi,\pi]^{8}}i\sum\limits_{l=0}^{7}\bm{\overline{e}}_{l}\sin(u_{l}){\left(\sum\limits_{l=0}^{7}\sin^{2}(u_{l})\right)^{-1}}e^{i\sum\limits_{l=0}^{7}u_{l}x_{l}}du.$

###### Proof.

We may assume $h=1.$ Define the discrete octonionic Cauchy-Fueter operator
$\bm{D}^{1}$ as

$\bm{D}^{1}=\sum\limits_{i=0}^{7}\bm{e}_{i}\partial_{l},$

and $\bm{\overline{D}}^{1}$ and its conjugate as

$\bm{\overline{D}}^{1}=\bm{e}_{0}\partial_{0}-\sum\limits_{i=1}^{7}\bm{e}_{i}\partial_{l}.$

A direct calculation yields

(4.7) $\Delta^{1}=\bm{D}^{1}\bm{\overline{D}^{1}}.$

With the definitions of $E^{1}$, $F$, and noting that

$\bm{\overline{D}}^{1}\Big{(}e^{i\sum\limits_{l=0}^{7}u_{l}x_{l}}\Big{)}=e^{i\sum\limits_{l=0}^{7}u_{l}x_{l}}\sum\limits_{l=0}^{7}i\bm{\overline{e}_{l}}\sin(u_{l}),$

we deduce

(4.8) $\bm{\overline{D}^{1}}F^{1}=\bm{E}^{1}.$

From (4.2),(4.7), it follows that

$\bm{D}^{1}\bm{E}^{1}=\bm{D}^{1}\bm{\overline{D}^{1}}F^{1}=\Delta^{1}F^{1}=\delta_{0}^{1}.$

This completes the proof. ∎

######  Lemma 7.

The fundamental solution $\bm{E}^{1}$ is an element of
$L^{2}(\mathbb{Z}^{8}).$

###### Proof.

Recall

$\bm{E}^{1}(x):=-\frac{1}{(2\pi)^{8}}\int_{[-\pi,\pi]^{8}}i\sum\limits_{l=0}^{7}\bm{\overline{e}}_{l}\sin(u_{l}){\left(\sum\limits_{l=0}^{7}\sin^{2}(u_{l})\right)^{-1}}e^{i\sum\limits_{l=0}^{7}u_{l}x_{l}}du.$

It implies $\bm{E}^{1}$ is the Fourier transform of

$i\sum\limits_{l=0}^{7}\bm{\overline{e}}_{l}\sin(u_{l}){\left(\sum\limits_{l=0}^{7}\sin^{2}(u_{l})\right)^{-1}}.$

We claim that this function is dominated by

$\frac{1}{\sqrt{\sum\limits_{l=0}^{7}(u_{l}-s_{l})^{2}}}$

near any of its singular points $(s_{0},s_{1},\cdots,s_{7}).$ Indeed, by
periodicity, we only need to consider the singular point $(0,0,0,0)$, where

$\left\|i\sum\limits_{l=0}^{7}\bm{\overline{e}}_{l}\sin(u_{l}){\left(\sum\limits_{l=0}^{7}\sin^{2}(u_{l})\right)^{-1}}\right\|=\frac{1}{\sqrt{\sum\limits_{l=0}^{7}\sin^{2}u_{l}}}\thicksim\frac{1}{\sqrt{\sum\limits_{l=0}^{7}u_{l}^{2}}}.$

Thus it belongs to $L^{2}([-\pi,\pi]^{4})$ and by Parseval’s identity we have
$\bm{E}^{1}\in L^{2}(\mathbb{Z}^{8}_{h}).$ ∎

######  Remark 3.

The integration expression of $\bm{E}^{1}$ from (4.6) shows that $x=0$ is not
a singular point of $\bm{E}^{1}$, contrasting the continuous case where

$\bm{E}(x)=\frac{1}{\omega_{7}}\frac{\bm{\bar{x}}}{|\bm{x}|^{8}}.$

Here $\omega_{7}=\frac{2\pi^{4}}{\Gamma(4)}$ is the surface area of the unit
sphere $S^{7}$.

Here’s a mathematically formatted version:

In the discrete context, the boundary integral kernel of $\bm{D}^{h}$ differs
from its continuous counterpart. Specifically, while the continuous version is
characterized by the fundamental solution $\bm{E}^{1}$ the discrete form is
described by

(4.9)
$\bm{K}^{h}(x,y):=-\frac{1}{2}\sum\limits_{i=0}^{7}\left(\bm{E}^{h}(he_{l}-x+y)n_{l}^{-}(x)+\bm{E}^{h}(-he_{l}-x+y)n_{l}^{+}(x)\right)\bm{e}_{l}.$

Given the octonionic nature of the terms within this kernel, multiplying
$\bm{K}^{h}(x,y)$ by an octonionic function, say $\bm{f}(y)$, presents
challenges due to the non-associativity of the octonions. For our purposes, we
adopt a specific multiplication method that respects this non-associativity:

(4.10)
$\bm{K}^{h}(x,y)*\bm{f}(x):=-\frac{1}{2}\sum\limits_{l=0}^{7}\left(\bm{E}^{h}(he_{l}-x+y)n_{l}^{-}(x)+\bm{E}^{h}(-he_{l}-x+y)n_{l}^{+}(x)\right)(\bm{e}_{l}\bm{f}(x))$

By using this approach, we can develop a Cauchy-Pompeiu formula for
$\bm{D}^{h}$ that mirrors its continuous counterpart in form, but compensates
with unique boundary and volume integral kernels.

######  Theorem 3 (Cauchy-Pompeiu formula).

Let $B$ be a bounded set in $\mathbb{Z}^{8}_{h}.$ Then for any function
$\bm{f}:\overline{B}\longrightarrow\mathbb{O},$ we have

(4.11) $\displaystyle\chi_{B}(y)\bm{f}(y)=\int_{\partial
B}\bm{K}^{h}(x,y)*\bm{f}(x)dS(x)+\int_{B}\bm{E}^{h}(y-x)(\bm{D}^{h}\bm{f})(x)dV^{h}(x).$

###### Proof.

Employing the discrete divergence theorem on the boundary integral, we get

(4.12) $\displaystyle\int_{\partial
B}\bm{K}^{h}(x,y)*\bm{f}(x)dS(x)=\int_{B}G^{h}(x,y)dV^{h}(x),$

where the function $G^{h}(x,y)$ is defined as

$G^{h}(x,y)=-\frac{1}{2}\sum\limits_{l=0}^{7}\Bigg{(}\partial_{l}^{+,h}\Big{(}\bm{E}^{h}(he_{l}-\cdot+y)(\bm{e}_{l}\bm{f}(\cdot))\Big{)}(x)+\partial_{l}^{-,h}\Big{(}\bm{E}^{h}(-he_{l}-\cdot+y)(\bm{e}_{l}\bm{f}(\cdot))\Big{)}(x)\Bigg{)}.$

Applying the Leibniz rule, we can evaluate the components of the sum as

$\displaystyle\partial_{l}^{+,h}\Big{(}\bm{E}^{h}(he_{l}-\cdot+y)(\bm{e}_{l}\bm{f}(\cdot))\Big{)}(x)$
$\displaystyle=\bm{E}^{h}(-x+y)(\bm{e}_{l}\partial_{l}^{+,h}\bm{f}(x))-(\partial_{l}^{-,h}\bm{E}^{h})(-x+y)(\bm{e}_{l}\bm{f}(x)),$
$\displaystyle\partial_{l}^{-,h}\Big{(}\bm{E}^{h}(-he_{l}-\cdot+y)(\bm{e}_{l}\bm{f}(\cdot))\Big{)}(x)$
$\displaystyle=-(\partial_{l}^{+,h}\bm{E}^{h})(-x+y)(\bm{e}_{l}\bm{f}(x))+\bm{E}^{h}(-x+y)(\bm{e}_{l}\partial_{l}^{-,h}\bm{f}(x)).$

By direct computation, this gives

$\displaystyle G^{h}(x,y)$
$\displaystyle=(\bm{E}^{h}\bm{D}^{h})(-x+y)\bm{f}(x)-\bm{E}^{h}(-x+y)(\bm{D}^{h}\bm{f}(x))-\sum\limits_{l=0}^{7}[(\partial_{l}^{h}\bm{E}^{h})(-x+y),\bm{e}_{l},\bm{f}(x)]$
$\displaystyle=\delta^{h}_{0}(-x+y)\bm{f}(x)-\bm{E}^{h}(-x+y)(\bm{D}^{h}\bm{f}(x))-\sum\limits_{l=0}^{7}[(\partial_{l}^{h}\bm{E}^{h})(-x+y),\bm{e}_{l},\bm{f}(x)].$

We claim that the last term vanishes, i.e.,

$\sum\limits_{l=0}^{7}[(\partial_{l}^{h}\bm{E}^{h})(-x+y),\bm{e}_{l},\bm{f}]=0.$

Given this claim, integrating over $B$ yields

$\displaystyle\int_{B}G^{h}(x,y)dV^{h}(x)=\bm{f}(y)\chi_{B}(y)-\int_{B}\bm{E}^{h}(-x+y)(\bm{D}^{h}\bm{f}(x))dV^{h}(x).$

Combining this with (4.12), we arrive at our conclusion.

To verify our claim, consider that

$\bm{E}^{h}(y-x)=\bm{\overline{D}}^{h}F^{h}(y-x).$

Here $F^{h}(y-x)$ is the fundamental solution of the discrete Laplacian
$\Delta^{h}$ from (4.1). It satisfies the relation

$F^{h}(-x+y)=\frac{1}{h^{7}}F^{1}(\frac{-x+y}{h})$

and it is real-valued. Thus

$\displaystyle\sum\limits_{l=0}^{7}[(\partial_{l}^{h}\bm{E}^{h})(-x+y),\bm{e}_{l},\bm{f}]$
$\displaystyle=\sum\limits_{l=0}^{7}\sum\limits_{t=0}^{7}[\partial_{l}\bm{\overline{e}}_{t}(\partial_{t}F^{h})(-x+y),\bm{e}_{l},\bm{f}]$
$\displaystyle=\sum\limits_{l=0}^{7}\sum\limits_{t=0}^{7}(\partial_{l}\partial_{t}F^{h})(-x+y)[\bm{\overline{e}}_{t},\bm{e}_{l},\bm{f}]$
$\displaystyle=0.$

This holds true as

$\partial_{l}^{h}\partial_{t}^{h}F=\partial_{t}^{h}\partial_{l}^{h}F$

and for each $t,l=0,\cdots,7$,

(4.13)
$[\bm{\overline{e}}_{l},\bm{e}_{t},\bm{f}]+[\bm{\overline{e}}_{t},\bm{e}_{l},\bm{f}]=0.$

∎

######  Remark 4.

Theorem 3 can be expanded to encompass general alternative algebras. However,
this is not applicable to Cayley-Dickson algebras, due to the need for the
alternative property of octonions as showcased in (4.13).

######  Theorem 4 (Cauchy integral theorem).

Let $B$ be a bounded set in $\mathbb{Z}^{8}_{h}.$ Then for any function
$\bm{f}\in M(\overline{B},\mathbb{O})$ such that $\bm{D}^{h}\bm{f}=0$ in $B$,
we have

$\chi_{B}(y)\bm{f}(y)=\int_{\partial B}\bm{K}^{h}(x,y)*\bm{f}(x)dS(x).$

###### Proof.

This follows directly from the discrete Cauchy-Pompeiu integral formula. ∎

The discrete Cauchy-Pompeiu integral formula prompts further exploration,
particularly the introduction of the discrete Cauchy-Bitsadze operator and the
discrete Teodorescu operator.

######  Definition 10.

Let $B$ be a bounded set in the lattice $\mathbb{Z}^{8}_{h}$.

1\. The discrete Cauchy-Bitsadze operator $\bm{C}^{h}_{\partial B}:M(\partial
B,\mathbb{O})\longrightarrow M(\mathbb{Z}^{8}_{h},\mathbb{O})$ is defined by

$\bm{C}^{h}_{\partial B}\bm{f}(y):=\int_{\partial
B}\bm{K}^{h}(x,y)*\bm{f}(x)dS(x).$

2\. The discrete Teodorescu operator
$\bm{T}^{h}_{B}:M(B,\mathbb{O})\longrightarrow
M(\mathbb{Z}^{8}_{h},\mathbb{O})$ is defined by

$\bm{T}^{h}_{B}\bm{f}(y):=\int_{B}\bm{E}^{h}(y-x)\bm{f}(x)dV^{h}(x).$

######  Theorem 5 (Inverse of Cauchy integral theorem).

Let $B$ be a bounded set in $\mathbb{Z}^{8}_{h},$ and $\bm{f}\in
M(\overline{B},\mathbb{O}).$ The following assertions hold:

  * •

The Cauchy type integral $\bm{C}^{h}_{\partial B}\bm{f}$ is regular both in
$B^{\circ}$ and in $(\mathbb{Z}^{8}_{h}\setminus B)^{\circ}.$

  * •

If there holds $\bm{f}(y)=\bm{C}^{h}_{\partial B}\bm{f}(y)$ for any $y\in B,$
then $\bm{f}$ is regular in $B^{\circ}$.

###### Proof.

Since $B$ is a bounded set in $\mathbb{Z}^{8}_{h}$, it is a finite subset so
that the integral by the Cauchy type integral is a finite sum. This means we
can calculate differential under the integral sign. Hence it suffices to show
that for any $x\in\partial B$ and any $y\in
B^{\circ}\cup(\mathbb{Z}^{8}_{h}\setminus B)^{\circ}$ we have

$\bm{D}^{h}\left(\bm{K}^{h}(x,\cdot)*\bm{f}(x)\right)(y)=0.$

Indeed, by the definition of the star product from (4.10), we have

$\displaystyle\quad\bm{D}^{h}\left(\bm{K}^{h}(x,\cdot)*\bm{f}(x)\right)(y)$
$\displaystyle=\bm{D}^{h}_{y}\left(-\frac{1}{2}\sum\limits_{l=0}^{7}\left(\bm{E}^{h}(he_{l}-x+y)n_{l}^{-}(x)+\bm{E}^{h}(-he_{l}-x+y)n_{l}^{+}(x)\right)(\bm{e}_{l}\bm{f}(x))\right)$
$\displaystyle=-\frac{1}{2}\left(\bm{D}^{h}_{y}\sum\limits_{l=0}^{7}\bm{E}^{h}(he_{l}-x+y)n_{l}^{-}(x)+\bm{E}^{h}(-he_{l}-x+y)n_{l}^{+}(x)\right)(\bm{e}_{l}\bm{f}(x))$
$\displaystyle+\frac{1}{2}[\bm{D}^{h}_{y},\sum\limits_{l=0}^{7}\bm{E}^{h}(he_{l}-x+y)n_{l}^{-}(x)+\bm{E}^{h}(-he_{l}-x+y)n_{l}^{+}(x),\bm{e}_{l}\bm{f}(x)]$
$\displaystyle=\frac{1}{2}[\bm{D}^{h}_{y},\sum\limits_{l=0}^{7}\bm{E}^{h}(he_{l}-x+y)n_{l}^{-}(x)+\bm{E}^{h}(-he_{l}-x+y)n_{l}^{+}(x),\bm{e}_{l}\bm{f}(x)].$

In the last equality, we used Lemma 6, which means for any $x\in\partial B$
and $y\in(\mathbb{Z}^{8}_{h}\setminus\partial B)^{\circ},$

$\displaystyle\bm{D}^{h}_{y}\bm{E}^{h}(he_{l}-x+y)=\delta^{h}_{0}(he_{l}-x+y)=0,$
$\displaystyle\bm{D}^{h}_{y}\bm{E}^{h}(-he_{l}-x+y)=\delta^{h}_{0}(-he_{l}-x+y)=0.$

Thus it remains to show

$[\bm{D}^{h}_{y},\sum\limits_{l=0}^{7}\bm{E}^{h}(he_{l}-x+y)n_{l}^{-}(x)+\bm{E}^{h}(-he_{l}-x+y)n_{l}^{+}(x),\bm{e}_{l}\bm{f}(x)]=0.$

We claim that for each $l=0,1\cdots,7,$

$\left[\bm{D}^{h}_{y},\bm{E}^{h}(he_{l}-x+y)n_{l}^{-}(x),\bm{e}_{l}\bm{f}(x)\right]=\left[\bm{D}^{h}_{y},\bm{E}^{h}(-he_{l}-x+y)n_{l}^{+}(x),\bm{e}_{l}\bm{f}(x)\right]=0.$

By (4.8),

$\bm{\overline{D}}^{h}F=\bm{E}^{h},$

where $F^{h}$ is scale, the fundamental solution of the discrete Laplacian
$\Delta^{h}$ given by

$F^{h}(x)=\frac{1}{h^{7}}F^{1}(\frac{x}{h}),$

see (4.1). We have

$\displaystyle\quad\left[\bm{D}^{h}_{y},\bm{E}^{h}(he_{l}-x+y)n_{l}^{-}(x),\bm{e}_{l}\bm{f}(x)\right]$
$\displaystyle=[\bm{D}^{h}_{y},(\bm{D}^{h}_{y}F^{h})(he_{l}-x+y)n_{l}^{-}(x),\bm{e}_{l}\bm{f}(x)]$
$\displaystyle=[\sum\limits_{i=0}^{7}\bm{e}_{i}\partial^{h}_{i},\sum\limits_{j=0}^{7}\bm{e}_{j}(\partial^{h}_{j}F^{h})(he_{l}-x+y)n_{l}^{-}(x),\bm{e}_{l}\bm{f}(x)]$
$\displaystyle=\sum\limits_{i,j=0}^{7}(\partial^{h}_{i}\partial^{h}_{j}F)(he_{l}-x+y)[\bm{e}_{i},\bm{e}_{j}n^{-}_{l}(x),\bm{e}_{l}\bm{f}(x)].$

Since $\partial^{h}_{i}\partial^{h}_{j}F=\partial^{h}_{j}\partial^{h}_{i}F$
and

$[\bm{e}_{i},\bm{e}_{j}n^{-}_{l}(x),\bm{e}_{l}\bm{f}(x)]+[\bm{e}_{j},\bm{e}_{i}n^{-}_{l}(x),\bm{e}_{l}\bm{f}(x)]=0,$

we get

$\left[\bm{D}^{h}_{y},\bm{E}^{h}(he_{l}-x+y)n_{l}^{-}(x),\bm{e}_{l}\bm{f}(x)\right]=0.$

Similarly,

$\left[\bm{D}^{h}_{y},\bm{E}^{h}(-he_{l}-x+y)n_{l}^{+}(x),\bm{e}_{l}\bm{f}(x)\right]=0.$

This finishes the proof. ∎

As in the continuous setting, the Teodorescu operator $\bm{T}^{h}_{B}$ is a
right inverse of the Cauchy-Fueter operator $\bm{D}^{h}$:

######  Theorem 6.

Let B be a bounded set in $\mathbb{Z}^{8}_{h}.$ Then for any $\bm{f}\in
M(B,\mathbb{O}),$ there holds

$\bm{D}^{h}(\bm{T}^{h}_{B}\bm{f})(y)=\chi_{B}(y)\bm{f}(y).$

###### Proof.

Since the discrete integral

$\bm{T}^{h}_{B}\bm{f}(y):=\int_{B}\bm{E}^{h}(y-x)\bm{f}(x)dV^{h}(x)$

over a bounded $B$ is actually a finite sum, we can calculate differential
under the integral sign. That is,

(4.14) $\displaystyle\bm{D}^{h}(\bm{T}^{h}_{B}\bm{f})(y)$
$\displaystyle=\int_{B}\bm{D}^{h}_{y}(\bm{E}^{h}(y-x)\bm{f}(x))dV^{h}(x)$
$\displaystyle=\int_{B}(\bm{D}^{h}_{y}\bm{E}^{h}(y-x))\bm{f}(x)+[\bm{D}^{h}_{y},\bm{E}(y-x),\bm{f}(x)]dV^{h}(x)$
$\displaystyle=\chi_{B}(y)\bm{f}(y)+\int_{B}[\bm{D}^{h}_{y},\bm{E}(y-x),\bm{f}(x)]dV^{h}(x).$

So it suffices to show

$[\bm{D}^{h}_{y},\bm{E}(y-x),\bm{f}(x)]=0.$

By (4.8),

$\bm{\overline{D}}^{h}F=\bm{E}^{h},$

where $F^{h}$ is scale, the fundamental solution of the discrete Laplacian
$\Delta^{h}$ given by

$F^{h}(x)=\frac{1}{h^{7}}F^{1}(\frac{x}{h}),$

see (4.1). We have that

$\displaystyle=[\bm{D}^{h}_{y},\bm{\overline{D}}^{h}_{y}F(y-x),\bm{f}(x)]$
$\displaystyle=[\sum\limits_{l=0}^{7}\bm{e}_{l}\partial^{h}_{y_{l}},\sum\limits_{m=0}^{7}\overline{\bm{e}}_{m}\partial^{h}_{y_{m}}F(y-x),\bm{f}(x)]$
$\displaystyle=\sum\limits_{l,m=0}^{7}\partial^{h}_{y_{l}}\partial^{h}_{y_{m}}F(y-x)[\bm{e}_{l},\overline{\bm{e}}_{m},\bm{f}(x)]$
$\displaystyle=0.$

In the last equality, we use the fact that

$\partial^{h}_{y_{l}}\partial^{h}_{y_{m}}F(y-x)=\partial^{h}_{y_{m}}\partial^{h}_{y_{l}}F(y-x)$

and

$[\bm{e}_{l},\overline{\bm{e}}_{m},\bm{f}(x)]+[\bm{e}_{m},\overline{\bm{e}}_{l},\bm{f}(x)]=0$

for any $l,m=0,1,\cdots,7.$

∎

##  5\. Estimation of the Discrete Cauchy Kernel

Let us consider the discrete Cauchy kernel in the lattice structure. We aim to
provide a quantitative estimate for this kernel, which proves to be pivotal in
the context of the discrete regular extension theory. Specifically, the
relevance of this estimate can be observed in Theorems 11 and 12.

Let $\mathcal{F}_{\mathbb{R}^{8}}$, $\mathcal{F}_{\mathbb{T}^{8}}$ be the
Fourier transforms on $\mathbb{R}^{8}$ and on $\mathbb{T}:=[-\pi,\pi]^{8}$
respectively, i.e.

$\displaystyle\mathcal{F}_{\mathbb{R}^{8}}g(v)$
$\displaystyle=\frac{1}{(2\pi)^{8}}\int_{\mathbb{R}^{8}}g(u)e^{iu\cdot v}du,$
$\displaystyle\mathcal{F}_{\mathbb{T}^{8}}g(v)$
$\displaystyle=\frac{1}{(2\pi)^{8}}\int_{\mathbb{T}^{8}}g(u)e^{iu\cdot v}du.$

It is known that the fundamental solution of the Dirac operator in octonions

$D=\sum_{l=0}^{7}e_{l}\partial_{l}$

is given by

$\bm{E}(x)=\dfrac{3}{\pi^{8}}\dfrac{\bm{\overline{x}}}{|x|^{8}}.$

For its discrete version, we know that the discrete Dirac operator

$D^{h}=\sum_{l=0}^{7}e_{l}\partial_{l}^{h}$

has the the fundamental solution $E^{h}$, given in (4.6). We call it the
discrete Cauchy kernel of octonions. Without loss of generality, we may assume
$h=1$.

Within the Fourier domain, we consider the discrete and continuous fundamental
solutions above, which take the form

(5.1) $\displaystyle\bm{S_{c}}$
$\displaystyle=\mathcal{F}_{\mathbb{R}^{8}}^{-1}\bm{E},$
$\displaystyle\bm{S_{d}}$
$\displaystyle=\mathcal{F}_{\mathbb{T}^{8}}^{-1}\bm{E}^{1}.$

By direct calculation, they can be expressed explicitly as

$\begin{split}\bm{S_{d}}(u)&=\dfrac{\sum_{l=0}^{7}\overline{\bm{e}_{l}}\,\sin
u_{l}}{i\sum_{l=0}^{7}\sin^{2}u_{l}},\\\
\bm{S_{c}}(u)&=\dfrac{\sum_{l=0}^{7}\overline{\bm{e}_{l}}\,u_{l}}{i|u|^{2}}.\end{split}$

This two functions exhibit distinct phenomena for their singularities. In
contrast to the classical case that $\bm{S_{c}}$ has its sole singular point
at $u=0$, $\bm{S_{d}}$ exhibits 24 singular points characterized by
$u=(u_{0},u_{1},\dots,u_{7})$ where $u_{i}\in\\{0,\pm\pi\\}$. An alternative
representation is

$u=\pi\sum_{l=0}^{7}\alpha_{l}e_{l},\quad\alpha_{l}\in\\{0,\pm 1\\}.$

We will provide an expression of the discrete Cauchy kernel in terms of the
continuous Cauchy kernel as well as some auxiliary functions introduced below.

1\. $\phi:\mathbb{R}^{8}\to\mathbb{R}$ is a smooth, real-valued cutoff
function, defined as

$\phi(u)=\begin{cases}1&\text{if
}u\in\left[-\frac{\pi}{8},\frac{\pi}{8}\right]^{8},\\\ 0&\text{if
}u\notin\left[\frac{\pi}{4},\frac{\pi}{4}\right]^{8}.\end{cases}$

Utilizing $\phi$, we can discern the singular points from the aforementioned
symbols, leading to auxiliary functions (2) and (3).

2\. $\Phi:\mathbb{R}^{8}\to\mathbb{R}$ is a modified discrete fundamental
solution, defined by

$\Phi(u)=(\phi(u)-1)\bm{S_{c}}(u).$

3\. We consider an error function

$\Theta(u)=\bm{S_{d}}(u)-\sum_{\begin{subarray}{c}\alpha_{l}\in\\{0,\pm
1\\},\\\ 0\leq l\leq
7\end{subarray}}\phi\left(u+\pi\sum_{l=0}^{7}\alpha_{l}e_{l}\right)\bm{S_{c}}\left(u+\pi\sum_{l=0}^{7}\alpha_{l}e_{l}\right)$

4\. Introducing a bounded weight function:

$\omega(x)=\prod_{l=0}^{7}\left(1+2\cos(\pi x_{l})\right)$

Now we can provide close relation between the discrete and continuous Cauchy
kernels.

######  Theorem 7.

Te fundamental solution $\bm{E}^{1}$ admits the following expansion:

(5.2)
$\bm{E}^{1}(x)=\omega(x)\bm{E}(x)+\frac{1}{|x|^{8}}\omega(x)\mathcal{F}_{\mathbb{R}^{8}}(\Delta^{4}\Phi)(x)+\frac{1}{|x|^{8}}\mathcal{F}_{\mathbb{T}^{8}}(\Delta^{4}\Theta)(x)$

for any $x\in\mathbb{Z}^{8}\setminus\\{0\\},$ In particular,

$\bm{E}^{1}(x)=\omega(x)\bm{E}(x)+O(|x|^{-8})$

as $|x|\rightarrow+\infty$.

The proof of Theorem 7 relies heavily on the remarkable work by Shivakumar and
Wong [32] concerning the asymptotic expansion of Fourier transforms. We denote

$L_{q}=2^{q+n/2}\Gamma\left(\dfrac{q+n}{2}\right)\left(\Gamma\left(\dfrac{-q}{2}\right)\right)^{-1}.$

######  Theorem 8 ([32]).

Let function f $\in C^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ admit an
expansion of a finite sum

$f(u)=\sum\limits_{p,q}c_{p,q}u^{p}|u|^{q}+\Phi(u),$

where

$q+n>0$

for all the real numbers q under the summation, $p=(p_{1},\cdots,p_{n})$ is a
multi-index of non-negative integers, $u^{p}=u_{1}^{p_{1}}\cdots
u_{n}^{p_{n}}$, and $\Phi\in C^{\infty}(\mathbb{R}^{n}).$ Then there holds

$\mathcal{F}_{\mathbb{R}^{n}}f(v)=\sum\limits_{p,q}c_{p,q}L_{q}\left(\frac{1}{i}\frac{\partial}{\partial
v_{1}}\right)^{p_{1}}\cdots\left(\frac{1}{i}\frac{\partial}{\partial
v_{n}}\right)^{p_{n}}(|v|^{-q-n})+\dfrac{(-1)^{m}}{|v|^{2m}}\mathcal{F}_{\mathbb{R}^{n}}(\Delta^{m}\Phi)(v)$

provided the following assumptions holds true:

(1) We assume

$2m-1\leqslant Q+n\leqslant 2m+1$

where

$Q=\mathop{\max}\limits_{p,q}(q+|p|)$

with the maximum taken over all the multi-indices $p$ and the real number $q$
under the summation.

(2) For any $j=0,1,...,m,$ assume that, as $u\rightarrow 0,$

$\Delta^{j}\Phi(u)=\begin{cases}O(|u|^{Q-2j+2}),&Q+n=2m-1,\\\ \\\
O(|u|^{Q-2j+1}),&Q+n\neq 2m-1.\end{cases}$

(3) There exists a constant $\rho>0$ such that, for any $j=0,1,...,m,$ the
limit

$\lim_{R\rightarrow+\infty}\int_{\rho<|u|<R}\Delta^{j}f(u)e^{iu\cdot v}du$

converges uniformly with respect to $v$ for sufficiently large $|v|$.

###### Proof of Theorem 7.

The expansion (5.2) can be derived by the following two claims:

claim 1:

$\bm{E}^{1}(x)=\omega(x)\mathcal{F}_{\mathbb{R}^{8}}(\phi\bm{S}_{c})(x)+\dfrac{1}{|x|^{8}}\mathcal{F}_{\mathbb{T}^{8}}(\Delta^{4}\Theta)(x),$

claim 2:

$\mathcal{F}_{\mathbb{R}^{8}}(\phi\bm{S}_{c})(x)=\bm{E}(x)+\dfrac{1}{|x|^{8}}\mathcal{F}_{\mathbb{R}^{8}}(\Delta^{4}\Theta)(x).$

We prove claim 1 first. By (5.1),
$\bm{E}^{1}=\mathcal{F}_{\mathbb{T}}^{8}\bm{S}_{d},$ which means

$\bm{E}^{1}=\mathcal{F}_{\mathbb{T}}^{8}\left(\sum\limits_{\begin{subarray}{c}\alpha_{l}=0,\pm
1,\\\ 0\leqslant l\leqslant
7\end{subarray}}\phi(u+\pi\sum\limits_{l=0}^{7}\alpha_{l}e_{l})\bm{S}_{c}(u+\pi\sum\limits_{l=0}^{7}\alpha_{l}e_{l})\right)+\mathcal{F}_{\mathbb{T}}^{8}\Theta.$

A direct calculation shows that the first summand on the right side equals
$\omega\mathcal{F}_{\mathbb{R}^{8}}(\phi\bm{S}_{c}).$ On the other hand, since
all the derivatives of $\Theta$ up to the eighth order belong to
$L^{1}([-\pi,\pi]^{8})$, we have

$\mathcal{F}_{\mathbb{T}^{8}}\Theta=\frac{1}{|x|^{8}}\mathcal{F}_{\mathbb{T}^{8}}(\Delta^{4}\Theta).$

Combining the results above yields claim 1.

Now we prove claim 2. In Theorem 8, we take

$\bm{\Phi}=(\phi-1)\bm{S}_{c},$

and

$\bm{f}(u)=(\phi\bm{S}_{c})(u)=\dfrac{\sum_{l=0}^{7}\overline{\bm{e}}_{l}\,u_{l}}{i|u|^{2}}+\bm{\Phi}(u),$

also take the parameters

$n=8,\quad m=4,\quad Q=-1,\quad|p|=1,\quad q=-2.$

Then we obtain

$\displaystyle\mathcal{F}_{\mathbb{R}^{8}}(\phi\bm{S}_{c})(x)$
$\displaystyle=\frac{1}{2\pi^{4}i}\left(\dfrac{1}{i}\sum\limits_{l=0}^{7}\overline{\bm{e}}_{l}\dfrac{\partial}{\partial
x_{l}}\right)|x|^{-6}+\frac{1}{|x|^{8}}\mathcal{F}_{\mathbb{R}^{8}}(\Delta^{4}\bm{\Phi})(x)$
$\displaystyle=\dfrac{3}{\pi^{4}}\dfrac{\bm{\overline{x}}}{|x|^{8}}+\frac{1}{|x|^{8}}\mathcal{F}_{\mathbb{R}^{8}}(\Delta^{4}\bm{\Phi})(x)$
$\displaystyle=\bm{E}(x)+\frac{1}{|x|^{8}}\mathcal{F}_{\mathbb{R}^{8}}(\Delta^{4}\bm{\Phi})(x).$

This finishes the proof. ∎

Theorem 7 gives an estimate for the discrete fundamental solution
$\bm{E}^{1}$. For a much more common estimate for this function, we stipulate
that $\bm{E}(0):=0$ to get the following result.

######  Theorem 9.

For any $h>0$ and $x\in\mathbb{Z}^{8}_{h}$, we have

(5.3)
$|\bm{E}^{h}(x)-\omega(h^{-1}x)\bm{E}(x)|\leqslant\dfrac{Ch}{|x|^{8}+h^{8}}.$

###### Proof.

First, we take $h=1.$ Then by Theorem 7, we have

$|\bm{E}^{1}(x)-\omega(x)\bm{E}(x)|\leqslant\dfrac{C_{0}}{|x|^{8}},\quad
x\in\mathbb{Z}^{8}\setminus\\{0\\}$

where

$C_{0}=\frac{1}{\pi^{8}}\int_{\mathbb{R}^{8}}|\Delta^{4}\Phi(u)|du+\frac{1}{(2\pi)^{8}}\int_{[-\pi,\pi]^{8}}|\Delta^{4}\Theta(u)|du.$

Furthermore,

$|\bm{E}^{1}(0)-\omega(0)\bm{E}(0)|=|\bm{E}^{1}(0)|\neq+\infty.$

Thus

(5.4) $|\bm{E}^{1}(x)-\omega(x)\bm{E}(x)|\leqslant\dfrac{C}{|x|^{8}+1}$

with $C=\max\\{|\bm{E}^{1}(0),2C_{0}\\}.$

By the definitions of the discrete and continuous fundamental solutions, we
have

$\bm{E}^{h}(x)=h^{-7}\bm{E}^{1}(h^{-1}x),\quad x\in\mathbb{Z}^{8}_{h},$

and

$\bm{E}(x)=h^{-7}\bm{E}(h^{-1}x),\quad x\in\mathbb{R}^{8}.$

Therefore,

$|\bm{E}^{h}(x)-\omega(h^{-1}\bm{E}(x))|=h^{-7}|\bm{E}^{1}(x^{\prime})-\omega(x^{\prime})\bm{E}(x^{\prime})|$

with $x^{\prime}=h^{-1}x.$ Hence (5.4) yields that

$|\bm{E}^{h}(x)-\omega(h^{-1}x)\bm{E}(x)|\leqslant\dfrac{Ch^{-7}}{|x^{\prime}|^{8}+1}=\dfrac{Ch}{|x|^{8}+h^{8}}.$

∎

##  6\. Sokhotski-Plemelj formula

This section delves into the Sokhotski-Plemelj formula, spotlighting operators
like the discrete Cauchy-Bitsadze and Teodorescu. These operators, tied to the
lattice’s boundaries, interlink functions on boundaries with those spanning
the lattice. Key properties, like the discrete Cauchy-Pompeiu formula, enable
function decomposition. Furthermore, the Sokhotski-Plemelj theorem and the
discrete Plemelj projections underscore the significance of boundary values in
discerning function behaviors, solidifying our understanding of functions on
lattices.

######  Definition 11.

For a bounded set $B$ in the lattice $\mathbb{Z}^{8}_{h}$, we define the
operators:

  * •

The _discrete Cauchy-Bitsadze operator_ $\bm{C}^{h}_{\partial B}:M(\partial
B,\mathbb{O})\to M(\mathbb{Z}^{8}_{h},\mathbb{O})$, given by

$\bm{C}^{h}_{\partial B}\bm{f}(y):=\int_{\partial
B}\bm{K}^{h}(x,y)\ast\bm{f}(x)\,dS(x).$

  * •

The _discrete Teodorescu operator_ $\bm{T}^{h}_{B}:M(B,\mathbb{O})\to
M(\mathbb{Z}^{8}_{h},\mathbb{O})$, defined as

$\bm{T}^{h}_{B}\bm{f}(y):=\int_{B}\bm{E}^{h}(y-x)\bm{f}(x)\,dV^{h}(x).$

In parallel with the continuous context, the Teodorescu operator
$\bm{T}^{h}_{B}$ serves as a right inverse to the Cauchy-Fueter operator
$\bm{D}^{h}$. Formally, for any $\bm{f}\in M(B,\mathbb{O})$,

$\bm{D}^{h}\bm{T}^{h}_{B}\bm{f}=\bm{f}.$

Leveraging these notations, the discrete Cauchy-Pompeiu formula (refer to
Theorem 3) can be articulated as follows:

(6.1) $\bm{C}^{h}_{\partial
B}\bm{f}+\bm{T}^{h}_{B}(\bm{D}^{h}\bm{f})=\chi_{B}\bm{f}.$

######  Definition 12.

Given a bounded domain $B$ in the lattice $\mathbb{Z}^{8}_{h}$, we define an
integral operator $\bm{S}^{h}_{\partial B}:M(\partial B,\mathbb{O})\to
M(\partial B,\mathbb{O})$ by

$\bm{S}^{h}_{\partial B}\bm{f}(y)=2\int_{\partial
B}\bm{K}^{h}(x,y)\ast(\bm{f}(x)-\bm{f}(y))\,dS(x)+\bm{f}(y).$

######  Notation 1.

We represent the inner and outer boundaries of $B$ as $\partial^{+}B$ and
$\partial^{-}B$ respectively, where

$\partial^{+}B=\partial B\cap B\quad\text{and}\quad\partial^{-}B=\partial
B\setminus B.$

######  Theorem 10 (Discrete Sokhotski-Plemelj formula).

Given a bounded set $B$ in $\mathbb{Z}^{8}_{h}$ and any function $\bm{f}\in
M(\partial B,\mathbb{O})$, the following relation holds:

$\bm{C}^{h}_{\partial
B}\bm{f}(y)=\frac{1}{2}\left(\pm\bm{f}(y)+\bm{S}^{h}_{\partial
B}\bm{f}(y)\right)\quad\forall y\in\partial^{\pm}B.$

###### Proof.

Using the definition of $\bm{S}^{h}_{\partial B}$ for any $y\in\partial B$:

$\displaystyle\bm{S}^{h}_{\partial B}\bm{f}(y)$
$\displaystyle=2\bm{C}^{h}_{\partial B}(\bm{f}-\bm{f}(y))(y)+\bm{f}(y)$
$\displaystyle=2\bm{C}^{h}_{\partial
B}\bm{f}(y)-2\bm{f}(y)\bm{C}^{h}_{\partial B}(1)(y)+\bm{f}(y).$

Rearranging gives:

(6.2) $\bm{C}^{h}_{\partial B}\bm{f}(y)-\frac{1}{2}\bm{S}^{h}_{\partial
B}\bm{f}(y)=\bm{f}(y)\left(\bm{C}^{h}_{\partial B}(1)(y)-\frac{1}{2}\right).$

With the choice $\bm{f}\equiv 1$ in (6.1), we determine:

$\bm{C}^{h}_{\partial B}(1)(y)=\chi_{B}(y)\quad\forall y\in\partial B.$

Further, for $y\in\partial B$, by definition:

$\chi_{B}(y)=\begin{cases}1,&\text{if }y\in\partial^{+}B,\\\ 0,&\text{if
}y\in\partial^{-}B.\end{cases}$

Incorporating this into (6.2) completes the proof. ∎

######  Definition 13.

We define the discrete Plemelj projections by

$\displaystyle\bm{P}^{h}_{\partial B}\bm{f}$
$\displaystyle=\frac{1}{2}(I+\bm{S}^{h}_{\partial B})\bm{f},$
$\displaystyle\bm{Q}^{h}_{\partial B}\bm{f}$
$\displaystyle=\frac{1}{2}(I-\bm{S}^{h}_{\partial B})\bm{f}.$

Using Theorem 10, the following representation holds:

(6.3) $\bm{C}^{h}_{\partial B}\bm{f}(y)=\begin{cases}\bm{P}^{h}_{\partial
B}\bm{f}(y),&\text{if }y\in\partial^{+}B,\\\ \bm{P}^{h}_{\partial
B}\bm{f}(y)-\bm{f}(y),&\text{if }y\in\partial^{-}B.\end{cases}$

This relation reflects the discontinuity between the inner and outer
boundaries.

######  Theorem 11 (Discrete regular extensions).

For a bounded subset $B$ of $\mathbb{Z}^{8}_{h}$ and any function
$\bm{f}:\partial B\rightarrow\mathbb{O}$, we have

  1. (${\mathrm{i}}$)

The projection $\bm{P}^{h}_{\partial B}\bm{f}$ extends into $B$ as a discrete
regular function.

  2. (${\mathrm{ii}}$)

The projection $\bm{Q}^{h}_{\partial B}\bm{f}$ extends into
$\mathbb{Z}^{8}_{h}\setminus B$ as a discrete regular function vanishing at
infinity.

###### Proof.

(i) Let the zero-extension of $\bm{f}$ over the space $\mathbb{Z}^{8}_{h}$
also be denoted by $\bm{f}$. From (6.1), we deduce

$\bm{C}^{h}_{\partial
B}\bm{f}(y)=\begin{cases}\bm{f}(y)-\bm{T}^{h}_{B}(\bm{D}^{h}\bm{f})(y),&y\in\partial^{+}B,\\\
-\bm{T}^{h}_{B}(\bm{D}^{h}\bm{f})(y),&y\in\partial^{-}B.\end{cases}$

Combining with (6.3), we find

$\bm{P}^{h}_{\partial
B}\bm{f}(y)=\bm{f}(y)-\bm{T}^{h}_{B}(\bm{D}^{h}\bm{f})(y),\qquad\forall\
y\in\partial B.$

Invoking Theorem 6, for any $\bm{g}\in M(B,\mathbb{O})$, we have:

$\bm{D}^{h}(\bm{T}^{h}_{B}\bm{g})(y)=\chi_{B}(y)\bm{g}(y).$

This implies that the function $\bm{f}(y)-\bm{T}^{h}_{B}(\bm{D}^{h}\bm{f})(y)$
mapping $\overline{B}\rightarrow\mathbb{O}$ is discrete regular. Hence,
$\bm{P}^{h}_{\partial B}\bm{f}(y)$ is extendable into $\overline{B}$ as a
discrete regular function.

(ii) Since

$\bm{P}^{h}_{\partial B}\bm{f}+\bm{Q}^{h}_{\partial B}\bm{f}=\bm{f},$

we obtain

$\bm{Q}^{h}_{\partial
B}\bm{f}(y)=\bm{T}^{h}_{B}(\bm{D}^{h}\bm{f})(y),\quad\forall\ y\in\partial B.$

The projection $\bm{Q}^{h}_{\partial B}\bm{f}(y)$ then corresponds to the
boundary values of the function

$\bm{T}^{h}_{B}(\bm{D}^{h}\bm{f}):\overline{\mathbb{Z}^{8}_{h}\setminus
B}\rightarrow\mathbb{O}.$

Using Theorem 6 again, this function is discrete regular.

To show that $\bm{T}^{h}_{B}(\bm{D}^{h}\bm{f})(y)$ vanishes at infinity,
consider its definition:

$\bm{T}^{h}_{B}(\bm{D}^{h}\bm{f})(y)=\int_{B}\bm{E}^{h}(y-x)(\bm{D}^{h}\bm{f})(y)dV^{h}(x).$

From Theorem 9, we can bound

$\displaystyle|\bm{E}^{h}(x)|$ $\displaystyle\leqslant$
$\displaystyle\left|\prod_{l=0}^{7}(1+2\mbox{cos}(h^{-1}\pi
x_{l}))\right||E(x)|+\frac{Ch}{|x|^{8}+h^{8}}$

so that

(6.4) $\displaystyle|\bm{E}^{h}(x)|$ $\displaystyle\leqslant$
$\displaystyle\frac{C}{|x|^{7}}+\frac{Ch}{|x|^{8}+h^{8}}.$

Here $C$ is a constant independent of $x$ and $h$.

Consequently,

$\begin{split}|\bm{T}^{h}_{B}(\bm{D}^{h}\bm{f})(y)|&\leqslant
V^{h}(B)\max_{B}|\bm{D}^{h}f|\max_{x\in B}|\bm{E}^{h}(y-x)|\\\ &\leqslant
V^{h}(B)\max_{B}|\bm{D}^{h}f|\left(\frac{C}{(|y|-\rho)^{7}}+\frac{Ch}{(|y|-\rho)^{8}+h^{8}}\right),\end{split}$

where

$\rho=\max_{x\in B}|x|.$

This verifies that $\bm{T}^{h}_{B}(\bm{D}^{h}\bm{f})(y)$ vanishes at infinity,
completing the proof. ∎

We now consider the Cauchy formula for discrete regular functions that vanish
at infinity, pivotal for characterizing their boundary values.

######  Lemma 8 (Cauchy formula for discrete regular functions).

Given a bounded subset $B$ of $\mathbb{Z}^{8}_{h}$ and a discrete regular
function $\bm{f}:\overline{\mathbb{Z}^{8}_{h}\setminus
B}\rightarrow\mathbb{O}$ that vanishes at infinity, the following relation
holds:

(6.5) $\displaystyle-\bm{C}^{h}_{\partial
B}\bm{f}(y)=\chi_{\mathbb{Z}^{8}_{h}\setminus B}(y)\bm{f}(y).$

###### Proof.

Let us extend $\bm{f}$ to the entirety of $\mathbb{Z}^{8}_{h}$ with zero
outside its original domain, and still denote it by $\bm{f}$. Define

$\Omega_{n}:=\left\\{y\in\mathbb{Z}^{8}_{h}:|y_{k}|\leqslant nh,\text{ for
}k=0,\dots,7\right\\}.$

Invoking the discrete Cauchy-Pompeiu formula from (6.1), we can write

$\bm{C}^{h}_{\partial\Omega_{n}}\bm{f}=\chi_{\Omega_{n}}\bm{f}-\bm{T}^{h}_{\Omega_{n}}(\bm{D}^{h}\bm{f})\quad\text{and}\quad\bm{C}^{h}_{\partial
B}\bm{f}=\chi_{B}\bm{f}-\bm{T}^{h}_{B}(\bm{D}^{h}\bm{f}).$

Subtracting these expressions leads to

(6.6) $\bm{C}^{h}_{\partial\Omega_{n}}\bm{f}-\bm{C}^{h}_{\partial
B}\bm{f}=\chi_{\Omega_{n}\setminus B}\bm{f}-\bm{T}^{h}_{\Omega_{n}\setminus
B}(\bm{D}^{h}\bm{f}),$

valid for sufficiently large values of $n$.

Since $\bm{D}^{h}\bm{f}$ vanishes on $\Omega_{n}\setminus B$, it follows that

$\bm{T}^{h}_{\Omega_{n}\setminus B}(\bm{D}^{h}\bm{f})=0,$

which simplifies (6.6) to

(6.7) $\bm{C}^{h}_{\partial\Omega_{n}}\bm{f}-\bm{C}^{h}_{\partial
B}\bm{f}=\chi_{\Omega_{n}\setminus B}\bm{f}.$

By letting $n$ go to infinity in (6.7) and recognizing that
$\chi_{\Omega_{n}\setminus B}$ converges to $\chi_{\mathbb{Z}^{8}_{h}\setminus
B}$, our task is now reduced to proving that

$\lim_{n\to\infty}\bm{C}^{h}_{\partial\Omega_{n}}\bm{f}(y)=0$

for every fixed $y$.

To approach this, note that the boundary $\partial\Omega_{n}$ is defined by

$\partial\Omega_{n}=\\{x\in\mathbb{Z}^{8}_{h}:x_{k}\in\\{nh,(n+1)h\\},\text{
and }|x_{j}|\leqslant nh\text{ for }j\neq k\\}.$

Hence, there exists a constant $C$, independent of $n$ and $h$, such that the
discrete boundary measure $S$ on $\partial\Omega_{n}$ satisfies

$S(\partial\Omega_{n})\leqslant Ch^{7}(n+1)^{7}.$

By definition, the Cauchy operator is given by

$\displaystyle\bm{C}^{h}_{\partial\Omega_{n}}\bm{f}(y)$
$\displaystyle=\int_{\partial\Omega_{n}}\bm{K}^{h}(x-y)\bm{f}(x)dS(x)$
$\displaystyle=\int_{\partial\Omega_{n}}-\frac{1}{2}\sum_{l=0}^{7}\left(\bm{E}^{h}(he_{l}-x+y)n_{l}^{-}(x)+\bm{E}^{h}(-he_{l}-x+y)n_{l}^{+}(x)\right)(\bm{e}_{l}\bm{f}(x))dS(x).$

We define the maximum absolute value of the integrand as $\bm{I}$. We thus
have

(6.8) $\displaystyle|\bm{C}^{h}_{\partial\Omega_{n}}\bm{f}(y)|\leqslant
S(\partial\Omega_{n})\max_{x\in\partial\Omega_{n}}|\bm{f}(x)|\bm{I}$
$\displaystyle\leqslant
Ch^{7}(n+1)^{7}\max_{x\in\partial\Omega_{n}}|\bm{f}(x)|\bm{I}.$

Since

$\sum_{l=0}^{7}(n_{l}^{-}+n_{l}^{+})=4,$

it follows that

$\bm{I}\leqslant\max_{x\in\partial\Omega_{n}}\sum_{l=0}^{7}|\bm{E}^{h}(he_{l}-x+y)|+|\bm{E}^{h}(-he_{l}-x+y)|.$

For a fixed $y$ and using (6.4), we deduce that

$\bm{I}\leqslant\left(\frac{16C}{(nh-|y|-h)^{7}}+\frac{16Ch}{(nh-|y|-h)^{8}+h^{8}}\right),$

which implies

(6.9) $\bm{I}=O(n^{-7}).$

Since $\bm{f}(y)$ vanishes at infinity, we have

(6.10) $\max_{x\in\partial\Omega_{n}}|\bm{f}(x)|=o(1).$

Inserting (6.9) and (6.10) into (6.8), it becomes clear that

$|\bm{C}^{h}_{\partial\Omega_{n}}|=o(1)$

as $n\to\infty$, completing the proof. ∎

######  Theorem 12.

Let $B$ be a bounded subset of $\mathbb{Z}^{8}_{h}$ and let $\bm{f}:\partial
B\to\mathbb{O}$ be any function. The following holds:

  1. (${\mathrm{i}}$)

$\bm{f}$ is the boundary value of a discrete regular function on $B$ if and
only if

$\bm{P}^{h}_{\partial B}\bm{f}(y)=\bm{f}(y),\qquad\forall\ y\in\partial B.$

  2. (${\mathrm{ii}}$)

$\bm{f}$ is the boundary value of a discrete regular function on
$\mathbb{Z}^{8}_{h}\setminus B$ that vanishes at infinity if and only if

$\bm{Q}^{h}_{\partial B}\bm{f}(y)=\bm{f}(y),\qquad\forall\ y\in\partial B.$

###### Proof.

The sufficiency is directly derived from Theorem 11, so we focus on
establishing the necessity.

(i) Suppose $\bm{f}:\partial B\to\mathbb{O}$ is the boundary value of a
discrete regular function $\bm{u}:B\to\mathbb{O}$. By equations (6.1) and
(6.3), we get

$\bm{C}^{h}_{\partial B}\bm{u}(y)=\bm{u}(y)\chi_{B}(y)$

and

$\bm{P}^{h}_{\partial B}\bm{f}(y)=\begin{cases}\bm{C}^{h}_{\partial
B}\bm{f}(y),&y\in\partial^{+}B,\\\ \bm{C}^{h}_{\partial
B}\bm{f}(y)+\bm{f}(y),&y\in\partial^{-}B.\end{cases}$

Combining the above, we deduce

$\bm{P}^{h}_{\partial B}\bm{f}(y)=\bm{f}(y),\qquad\forall\ y\in\partial B.$

(ii) If $\bm{f}:\partial B\to\mathbb{O}$ represents the boundary value of a
discrete regular function $\bm{u}:\overline{\mathbb{Z}^{8}_{h}\setminus
B}\to\mathbb{O}$ vanishing at infinity, then Lemma 8 states

$-\bm{C}^{h}_{\partial
B}\bm{f}(y)=\begin{cases}\bm{u}(y),&y\in\mathbb{Z}^{8}_{h}\setminus B,\\\
0,&y\in B.\end{cases}$

Furthermore, from Theorem 10, we have

$\bm{Q}^{h}_{\partial B}\bm{f}(y)=\begin{cases}-\bm{C}^{h}_{\partial
B}\bm{f}(y),&y\in\partial^{-}B,\\\ \bm{f}(y)-\bm{C}^{h}_{\partial
B}\bm{f}(y),&y\in\partial^{+}B.\end{cases}$

This results in $\bm{Q}^{h}_{\partial B}\bm{f}(y)=\bm{f}(y)$ for any
$y\in\partial B$, completing the proof. ∎

######  Corollary 2.

The following algebraic relations hold:

$\displaystyle(\bm{P}^{h}_{\partial B})^{2}$
$\displaystyle=\bm{P}^{h}_{\partial B},$ $\displaystyle(\bm{Q}^{h}_{\partial
B})^{2}$ $\displaystyle=\bm{Q}^{h}_{\partial B},$
$\displaystyle\bm{P}^{h}_{\partial B}\bm{Q}^{h}_{\partial B}$
$\displaystyle=\bm{Q}^{h}_{\partial B}\bm{P}^{h}_{\partial B}=0.$

###### Proof.

Considering $\bm{P}^{h}_{\partial B}+\bm{Q}^{h}_{\partial B}=I$, it is
sufficient to establish the first relation. For any $\bm{f}\in M(\partial
B,\mathbb{O})$, it follows from Theorem 11 that $\bm{P}^{h}_{\partial
B}\bm{f}$ represents the boundary values of a discrete regular function on
$B$. Therefore, from Theorem 12,

$(\bm{P}^{h}_{\partial B})^{2}\bm{f}=\bm{P}^{h}_{\partial B}\bm{f}.$

∎

##  7\. Convergence of Discrete Regular Octonionic Functions

In this section, we confirm that a continuous octonionic function is regular
precisely when it is a scale limit of discrete regular functions. In the
context of the complex case, as referenced in [11], this phenomenon is
underscored through the uniform stability of Dirichlet problems related to
discrete Laplace equations. Our approach is based on the methodology
introduced for the quaternionic scenario in [29].

###  7.1. Discrete Approximations of Domains in $\mathbb{R}^{8}$

######  Definition 14.

For a bounded domain $B$ within $\mathbb{R}^{8}$, a collection of sets
$B^{h}\subset\mathbb{Z}_{h}^{8}$ with $h>0$ is called to converge to $B$,
expressed as

$\lim_{h\mapsto 0}B^{h}=B,$

when the ensuing conditions are fulfilled:

(7.1) $\displaystyle\lim_{h\to 0}\max_{\alpha\in\partial
B}\min_{\beta\in\partial B^{h}}\|\alpha-\beta\|=0,$ $\displaystyle\lim_{h\to
0}\max_{\alpha\in\partial B^{h}}\min_{\beta\in\partial B}\|\alpha-\beta\|=0,$
$\displaystyle\lim_{h\to 0}\max_{\alpha\in\overline{B}}\min_{\beta\in
B^{h}}\|\alpha-\beta\|=0,$ $\displaystyle\lim_{h\to 0}\max_{\alpha\in
B^{h}}\min_{\beta\in\overline{B}}\|\alpha-\beta\|=0.$

Furthermore, if $B^{h}$ converge to $B$ and

$V^{h}(B^{h}\setminus B)=O(h^{2}),$

then the collection of sets $B^{h}$ is said to converge to $B$ at a rate of
$O(h^{2})$, symbolized as

$B=B^{h}+O(h^{2}).$

We can demonstrate that any bounded domain in $\mathbb{R}^{8}$ can be
approximated by discrete sets at a rate of $O(h^{2})$.

######  Theorem 13.

For a bounded domain $B$ in $\mathbb{R}^{8}$, the sets

$B^{h}\equiv(B\cap h\mathbb{Z}^{d})^{\circ}$

converge to $B$ at a rate of $O(h^{2})$.

###### Proof.

Let us confirm the four convergence identities from Definition 14.

(i) First we verify the fourth identity. Since $B^{h}$ is contained in $B$, we
have

$\min_{\beta\in\overline{B}}\|\alpha-\beta\|=0,\quad\forall\,\alpha\in B^{h}$

so that

$\lim_{h\mapsto 0}\max_{\alpha\in
B^{h}}\min_{\beta\in\overline{B}}\|\alpha-\beta\|=0.$

(ii) To verify the third identity, we aim to show that given any
$\varepsilon>0$, there exists a $\delta>0$ ensuring

$\overline{B}\subset B^{h}+B(0,2\varepsilon)$

for any $h\leqslant\delta$.

For any given $\varepsilon>0$ and $x\in\overline{B}$ one can find $\delta>0$
and $x^{\prime}\in B$ such that

$B(x^{\prime},\delta)\subset\subset B\cap B(x,\varepsilon).$

Consequently,

$B(x^{\prime},\delta)\cap\mathbb{Z}^{8}_{h}\subset(B\cap\mathbb{Z}^{8}_{h})^{\circ}=B^{h}$

for sufficiently small $h$, implying

$B(x^{\prime},\delta)\cap
B^{h}=B(x^{\prime},\delta)\cap\mathbb{Z}^{d}_{h}\neq\emptyset$

for sufficiently small $h$. From the fact

$B(x,\varepsilon)\cap B^{h}\supset B(x^{\prime},\delta)\cap B^{h},$

it follows that

(7.2) $B(x,\varepsilon)\cap B^{h}\neq\emptyset,\quad\forall h\ll 1.$

Due to the compactness of $B$, we can find a sequence $x_{1},\cdots,x_{n}$
such that

$\overline{B}\subset\bigcup\limits_{l=1}^{n}B(x_{l},\varepsilon).$

As we have proved for each $x_{l}$ there exists $\eta_{l}>0$ such that

$B(x_{l},\varepsilon)\cap B^{h}\neq\emptyset$

for any $h<\eta_{l}.$ Hence for any $h<\min\limits_{1\leqslant l\leqslant
n}\\{\eta_{l}\\},$ we have

$\overline{B}\subset B^{h}+B(0,2\varepsilon).$

(iii) Now we come to prove the second identity. It suffices to prove that

$\min_{y\in\partial B}\|x-y\|\leqslant 2h$

for any $x\in\partial B^{h}$.

Case 1: $x\in\partial^{+}B^{h}:=\partial B^{h}\cap B^{h}.$

For any $y\in\partial^{+}(B\cap\mathbb{Z}^{8}_{h}),$ according to the
definition of $\partial^{+}(B\cap\mathbb{Z}^{8}_{h}),$ we have $y\in B$ and
there is a point $y^{\prime}\in Z^{d}_{h}\setminus B$ such that
$\|y^{\prime}-y\|=h.$ The fact that $y\in B$ and $y^{\prime}\notin B$ leads to
the conclusion that

$\mathop{\min}\limits_{\beta\in\partial B}\|y-\beta\|\leqslant h$

for all $y\in\partial^{+}(B\cap\mathbb{Z}^{8}_{h})$.

For any $x\in\partial^{+}B^{h},$ there exists $y\in N(x)$ such that $y\notin
B^{h}$. Since $B^{h}$ is the discrete interior of $B\cap\mathbb{Z}^{8}_{h},$
we have $y\in\partial^{+}(B\cap\mathbb{Z}^{8}_{h}),$ which implies

$\min_{\beta\in\partial B}\|x-\beta\|\leqslant\|x-y\|+\min_{\beta\in\partial
B}\|y-\beta\|\leqslant 2h.$

Case 2:  $x\in\partial^{-}B^{h}:=\partial B^{h}\setminus B^{h}.$

From the fact that $B^{h}=(B\cap\mathbb{Z}^{8}_{h})^{\circ},$ it follows that

$x\in\partial^{+}(B\cap\mathbb{Z}^{8}_{h}).$

As shown above in case 1, we find

$\min_{\beta\in\partial B}\|x-\beta\|\leqslant h.$

(iv) At the final step, we come to verify the first identity

$\lim_{h\mapsto 0}\max_{\alpha\in\partial B}\min_{\beta\in\partial
B^{h}}\|\alpha-\beta\|=0.$

Since $\partial B$ is compact, it is sufficient to prove that for any
$\varepsilon>0$ and $x\in\partial B$, there exists $\delta>0$ such that

$\min_{\beta\in\partial B^{h}}\|x-\beta\|<\varepsilon$

for any $h<\delta.$

Indeed, for any given $\varepsilon>0$ and $x\in\partial B$, as shown in (7.2),
there exists $\eta>0$ such that

$B(x,\varepsilon)\cap B^{h}\neq\emptyset$

for any $h<\eta.$

On the other hand, because $B(x,\varepsilon)\setminus B$ is open, there exists
$\eta^{\prime}>0$ such that

$B(x,\varepsilon)\cap\mathbb{Z}^{8}_{h}\setminus\overline{B}\neq\emptyset$

for all $h<\eta^{\prime}$.

When $h<\min\\{\eta,\eta^{\prime}\\},$ we can thus find a sequence

$\\{x_{l}\\}_{l=1}^{n}\in\mathbb{Z}^{8}_{h}\cap B(x,\varepsilon)$

such that

$x_{1}\in B^{h},\quad x_{n}\in\mathbb{Z}^{8}_{h}\setminus B^{h}$

and

$\|x_{l}-x_{l+1}\|=h$

for all $1\leqslant l\leqslant n-1$.

This completes the proof.

∎

###  7.2. Convergence of Scaling Limits

In this subsection, we shall provide our main result about the convergence of
scaling limits. To this end we need two auxiliary lemmas below.

######  Lemma 9.

Let $B$ is a bounded open set in $\mathbb{R}^{8}$ and assume $\bm{f}\in
C^{3}(\mathbb{R}^{8})$ is regular on $B$. Then we have

$\max_{B}\|\bm{D^{h}f}\|=O(h^{2}).$

###### Proof.

Since $\bm{D}\bm{f}=0,$ we have

$\displaystyle\bm{D^{h}f}(x)$
$\displaystyle=\bm{D}^{h}\bm{f}(x)-\bm{D}\bm{f}(x)$
$\displaystyle=\sum\limits_{l=0}^{7}\bm{e}_{l}\frac{\bm{f}(x+he_{l})-\bm{f}(x-he_{l})-\partial_{x_{l}}\bm{f}(x)2h}{2h}.$

To deal with the last fractions, we need the following observation. For any
$g\in C^{3}(\mathbb{R}^{8},\mathbb{R})$, we have

$\displaystyle\frac{g(x+h)-g(x-h)-\partial_{x}g(x)2h}{2h}$
$\displaystyle=\frac{\partial_{x}g(x+h^{\prime})+\partial_{x}g(x-h^{\prime})-\partial_{x}g(x)2}{2}$
$\displaystyle=\frac{(\partial_{x}^{2}g(x+h^{\prime\prime})-\partial_{x}^{2}g(x-h^{\prime\prime}))h^{\prime}}{2}$
$\displaystyle=\partial_{x}^{3}(x+\lambda)h^{\prime}h^{\prime\prime}$

with $0<h^{\prime\prime}<h^{\prime}<h$ and
$\lambda\in(-h^{\prime\prime},+h^{\prime\prime}).$

Given the previous analysis, the conclusion is evident. ∎

######  Lemma 10.

Let $B$ be a bounded open set in $\mathbb{R}^{8}$. If $B^{h}\subset B$ for any
$h>0$, then

$\int_{B^{h}}1dV^{h}=O(1).$

###### Proof.

Since $B$ is bounded we may assume $B\subset\subset B(0,R).$ Therefore, we
have $B^{h}\subset B(0,R)$ so that

$\int_{B^{h}}1dV^{h}\leqslant\int_{\mathbb{Z}^{8}_{h}}\chi_{B(0,R)}dV^{h}.$

The last integral is identical to a Riemann sum of the integral
$\int_{B(0,R)}1dV$ where $dV$ is the Lebesgue measure on $\mathbb{R}^{8}$, so
that

$\lim_{h\mapsto 0+}\int_{\mathbb{Z}^{8}_{h}}\chi_{B(0,R)}dV^{h}=\pi R^{2}.$

Thus we come to the conclusion that

$\int_{B^{h}}1dV^{h}=O(1).$

∎

Now we come to the first main theorem in this section.

######  Theorem 14.

Suppose that $B$ is a bounded domain with $C^{3}$-boundary in $\mathbb{R}^{8}$
and

$B=B^{h}+O(h^{2}).$

Then for any regular function $\bm{f}\in C^{3}(\overline{B},\mathbb{O}),$
there exists a sequence of discrete regular functions
$\bm{f}^{h}:B^{h}\longrightarrow\mathbb{O}$ that converges to $f$ in the sense
that

$\lim_{h\mapsto}\max_{B^{h}\cap B}\|\bm{f}-\bm{f}^{h}\|=0.$

###### Proof.

Consider the discrete function $\bm{f}^{h}$ given by

(7.3) $\bm{f}^{h}(x):=\int_{\partial
B}\bm{K}^{h}(y,x)*\bm{\tilde{f}}(y)dS^{h}(y),$

where $\bm{\tilde{f}}$ is a $C^{3}_{0}$-extension of $\bm{f}$. By Theorem 5,
we know that $\bm{f}^{h}$ is discrete regular in $(B^{h})^{\circ}$. It
suffices to prove that the scaling limit of $\bm{f}^{h}$ is exactly $\bm{f}$.

Applying the discrete Cauchy-Pompeiu formula in Theorem 3 to the function
$\bm{\tilde{f}},$ we obtain

$\bm{f}(x)-\bm{f}^{h}(x)=\int_{B^{h}}\bm{E}^{h}(x-y)\left(\bm{D}^{h}\bm{\tilde{f}}(y)\right)dV^{h}(y)$

for any $x\in B^{h}\cap B.$ Consequently,

$\displaystyle\|\bm{f}(x)-\bm{\tilde{f}}(x)\|\leqslant$
$\displaystyle\max_{B^{h}\cap B}\|\bm{D}^{h}\bm{\tilde{f}}\|\int_{B^{h}\cap
B}\|\bm{E}^{h}(x-y)\|dV^{h}(y)$ $\displaystyle+\max_{B^{h}\setminus
B}\|\bm{D}^{h}\bm{\tilde{f}}\|\int_{B^{h}\cap B}\|\bm{E}^{h}(x-y)\|dV^{h}(y)$
$\displaystyle=$ $\displaystyle I_{1}+I_{2}.$

Observing the first term $I_{1}$, we have

$I_{1}\leqslant\max_{B^{h}\cap B}\|\bm{D}^{h}\|\left(\int_{B^{h}\cap
B}1dV^{h}\right)^{\frac{1}{2}}\left(\int_{B^{h}\cap
B}\|\bm{E}^{h}(x-y)\|^{2}dV^{h}(y)\right)^{\frac{1}{2}}.$

From the fact that $dV^{h}$ is the Haar measure of $Z^{8}_{h}$, we find

$\displaystyle\int_{B^{h}}\|\bm{E}(x-y)\|^{2}dV^{h}(y)$
$\displaystyle\leqslant\int_{\mathbb{Z}^{8}_{h}}\|\bm{E}(y)\|^{2}dV^{h}(y)$
$\displaystyle=\sum\limits_{y\in\mathbb{Z}^{8}_{h}}\|\frac{1}{h^{3}}\bm{E}(\frac{y}{h})\|^{2}h^{4}$
$\displaystyle=\frac{1}{h^{2}}\sum\limits_{x\in\mathbb{Z}^{8}}\|\bm{E}(x)\|$
$\displaystyle=\frac{1}{h^{2}}\int_{\mathbb{Z}^{8}}\|\bm{E}(x)\|dV$
$\displaystyle=O(h^{-2}).$

Utilizing Lemma 7, we note that $\bm{E}$ is an element of
$L^{2}(\mathbb{Z}^{8})$.

By Lemmas 9 and 10, we have

$\displaystyle\max_{B^{h}\cap B}\|\bm{D}^{h}\bm{\tilde{f}}\|$
$\displaystyle=O(h^{2}),$ $\displaystyle\int_{B^{h}\cap B}1dV^{h}$
$\displaystyle=O(1).$

From the preceding three evaluations, we derive $I_{1}\leqslant
O(h^{2})O(1)O(h^{-1})=O(h).$

According to the assumption that $B=B^{h}+O(h^{2})$, we have

$V^{h}(B^{h}\setminus B)=O(h^{2}).$

In addition, by the similar method used in the proof of Lemma 6.1, we can
prove that

$\max_{B^{h}\setminus B}\|\bm{D}^{h}\bm{\tilde{f}}\|=o(1).$

Indeed, as a direct result of the mean value theorem, we have

$\max_{B^{h}\cap
B}\|\bm{D}^{h}\bm{\tilde{f}}-\bm{D}\bm{\tilde{f}}\|=O(h^{2}),$

and from the facts that $B^{h}$ converges to $B$ and $\bm{D}\bm{\tilde{f}}=0$
on B, we can deduce that

$\max_{B^{h}}\|\bm{D}\bm{\tilde{f}}\|=o(1).$

Therefore,

$\max_{B^{h}\setminus B}\|\bm{D}^{h}\bm{\tilde{f}}\|=o(1).$

In the same way as we deal with $I_{1}$, we can also obtain

$\displaystyle I_{2}$ $\displaystyle\leqslant\max_{B^{h}\setminus
B}\|\bm{D}^{h}\bm{\tilde{f}}\|(V^{h}(B^{h}\setminus
B))^{\frac{1}{2}}\left(\int_{B^{h}\setminus
B}\|\bm{E}^{h}(x-y)\|^{2}dV^{h}(y)\right)^{\frac{1}{2}}$
$\displaystyle=o(1)O(h)O(h^{-1})=o(1).$

This completes the proof.

∎

Next, we demonstrate that when the scaling limit of discrete regular functions
is present, it is indeed regular.

######  Theorem 15.

Suppose $B$ is a bounded domain in $\mathbb{R}^{8}$ and $B^{h}\subset
Z^{8}_{h}$ converges to $B$. If a function $\bm{f}\in C^{1}(B)$ is the scaling
limit of discrete regular functions
$\bm{f}^{h}:B^{h}\longrightarrow\mathbb{O},$ i.e.,

$\lim_{h\mapsto 0+}\max_{B^{h}\cap B}\|\bm{f}-\bm{f}^{h}\|=0,$

then $\bm{f}$ is regular on $B$.

The proof of Theorem 15 relies on the next two lemmas.

######  Lemma 11.

Let $B$ be a bounded open set in $\mathbb{R}^{8}$ and assume
$B^{h}\subset\mathbb{Z}^{8}_{h}$ convergent to $B$. Then for any
$U\subset\subset B,$ there exists $\delta>0$ such that

$U\cap Z^{8}_{h}\subset B^{h}$

whenever $h<\delta.$

###### Proof.

Without loss of generality, we assume that $U=B(x_{0},R)\subset\subset B.$

From the assumption that $B^{h}$ converges to $B$,we have

$\lim_{h\mapsto 0}\max_{\alpha\in\overline{B}}\min_{\beta\in B^{h}}\|x-y\|=0,$

which means that, for any given $\varepsilon>0,$

$x_{0}\in\overline{B}\subset B^{h}+B(0,\varepsilon),$

when $h$ is sufficiently small. In particular, by taking $\varepsilon=R,$ we
have

$x_{0}\in B^{h}+B(0,R),$

i.e., $(x_{0}-B(0,R))\cap B^{h}\neq\emptyset,$ which means $U\cap
B^{h}\neq\emptyset.$

On the other hand, we denote

$d=\inf_{\alpha\in U,\beta\in\partial B}\|\alpha-\beta\|.$

From the assumption that $B^{h}$ converges to $B$, it follows that

$\lim_{h\mapsto 0+}\max_{\alpha\in\partial B^{h}}\min_{\beta\in\partial
B}\|\alpha-\beta\|=0.$

For any $h$ sufficiently small, we thus have

$\partial B^{h}\subset\partial B+B(0,d)$

so that

$\partial B^{h}\cap U\subset(\partial B+B(0,d))\cap U.$

Since $(\partial B+B(0,d))\cap U=\emptyset,$ we obtain

$\partial B^{h}\cap U=\emptyset.$

We have now established that for sufficiently small $h$ both conditions

$U\cap B^{h}\neq\emptyset,\qquad U\cap\partial B^{h}=\emptyset$

hold true.

Now we have proved that, when $h$ is sufficiently small,

$U\cap B^{h}\neq\emptyset$

and

$U\cap\partial B^{h}=\emptyset$

are both valid.

Based on these facts, we come to show that

$U\cap\mathbb{Z}^{8}_{h}\subset B^{h}$

for all sufficiently small $h$.

Assume this is not valid, then there exists $h^{*}$ such that

$U\cap B^{h^{*}}\neq\emptyset,\quad U\cap\partial B^{h^{*}}=\emptyset,\quad
U\cap\mathbb{Z}^{8}_{h}\nsubseteq B^{h^{*}}.$

From the fact that

$\alpha,\beta\in U\cap\mathbb{Z}^{8}_{h^{*}}=B(0,R)\cap\mathbb{Z}^{8}_{h}$

and that $B(x_{0},R)\cap\mathbb{Z}^{8}_{h}$ is discrete connected in the grid
$Z^{8}_{h^{*}}$, we can derive that there exists a sequence

$\\{x_{k}\\}_{k=1}^{m}\subset U\cap Z^{8}_{h^{*}}$

such that

$x_{1}=\alpha,\quad x_{m}=\beta,\quad\|x_{k+1}-x_{k}\|=h$

for each $k=1,2,\cdots,m-1.$

Notice that

$\alpha\in B^{h^{*}},\qquad\beta\notin B^{h^{*}},$

one can find $k^{*}$ with $1\leqslant k^{*}\leqslant m-1$ such that

$x_{k^{*}}\in B^{h^{*}},\quad x_{k^{*}+1}\notin B^{h^{*}}.$

According to the definition of discrete boundaries, $x_{k^{*}}$ and
$x_{k^{*}+1}$ are both in $\partial B^{h^{*}}$ . It violates the assumption
that $\partial B^{h^{*}}\cap U=\emptyset$. The proof is complete. ∎

######  Lemma 12.

Let $B$ be a bounded open set in $\mathbb{R}^{8}$ and $B^{h}\subset Z^{8}_{h}$
convergent to $B$. Then we have

$\lim_{h\mapsto 0+}\int_{B^{h}\cap B}\bm{f}dV^{h}=\int_{B}\bm{f}dV$

for any $\bm{f}\in C_{0}(B,\mathbb{O}).$

###### Proof.

Let’s consider any function $\bm{f}$ from the set $C_{0}(B)$. We can pinpoint
a set $U$, which is the finite union of balls, fulfilling the condition:

(7.4) $\mbox{supp}\bm{f}\subset U\subset\subset B.$

From the evidence provided in Lemma 11, when $h$ is sufficiently small, it is
clear that

$U\cap\mathbb{Z}^{8}_{h}\subset B^{h}\subset\mathbb{Z}_{h}^{8},$

which implies the equality $B^{h}\cap U=U\cap\mathbb{Z}^{8}_{h}.$

Utilizing both the above relation and (7.4), we can deduce

$\int_{B^{h}\cap B}\bm{f}dV^{h}=\int_{B^{h}\cap
U}\bm{f}dV^{h}=\int_{U\cap\mathbb{Z}^{8}_{h}}\bm{f}dV^{h}.$

It is worth noting that the latter integral serves as a Riemann sum for the
Riemann integral $\int_{U}\bm{f}dV$. Hence we obtain

$\lim_{h\mapsto 0+}\int_{B^{h}\cap B}\bm{f}dV^{h}=\lim_{h\mapsto
0+}\int_{U\cap\mathbb{Z}^{8}_{h}}\bm{f}dV^{h}=\int_{U}\bm{f}dV=\int_{B}\bm{f}dV.$

This concludes our proof. ∎

We now turn our attention to proving Theorem 15.

###### Proof of Theorem 15.

Assume that a sequence of discrete regular functions
$\bm{f}^{h}:B^{h}\longrightarrow\mathbb{O}$ converges to a function $\bm{f}\in
C^{1}(B,\mathbb{O}).$ We come to prove that $\bm{f}$ is regular on $B$.

We only need to verify that $\bm{Df}=0$ holds true in the sense of
distribution, i.e.,

(7.5) $\int_{B}(\phi\bm{D})\bm{f}dV=0$

for any $\phi\in C_{0}^{\infty}(B,\mathbb{R}).$

We emphasize that it is sufficient where test functions $\phi$ can be chosen
to be $\mathbb{R}$-valued rather than $\mathbb{O}$-valued. Moreover,
associators can be avoided in this manner.

Actually, if (7.5) holds for any $\phi\in C_{0}^{\infty}(B,\mathbb{R})$. Then
we have

(7.6) $0=\int_{B}(\phi\bm{D})\bm{f}dV=-\int_{B}\phi(\bm{Df})dV.$

Denote

$\bm{Df}=\sum\limits_{i=0}^{7}g_{i}\bm{e}_{i},$

where $g_{i}\in C_{0}(B,\mathbb{R}).$ Then (7.6) implies that for any $\phi\in
C_{0}^{\infty}(B,\mathbb{R})$,

(7.7) $\int_{B}\phi g_{i}=0,\quad i=0,\cdots,7.$

Thus $g_{i}=0$ if we let $\phi=g_{i}$ in (7.7) for each $i=0,\cdots,7.$

Now we prove (7.5). As shown in Lemma 12, we have

$\lim_{h\mapsto 0}\int_{B^{h}\cap
B}(\phi\bm{D})\bm{f}dV^{h}=\int_{B}(\psi\bm{D})\bm{f}dV.$

Hence it suffices to prove

$\lim_{h\mapsto 0+}\int_{B^{h}\cap B}(\phi\bm{D})\bm{f}dV^{h}=0.$

We now split the last integral into three parts

$\int_{B^{h}\cap B}(\phi\bm{D})\bm{f}dV^{h}=I^{h}_{1}+I^{h}_{2}+I^{h}_{3},$

where

(7.8) $\displaystyle\bm{I}^{h}_{1}$ $\displaystyle=\int_{B^{h}\cap
B}(\phi\bm{D})\bm{f}dV^{h},$ $\displaystyle\bm{I}^{h}_{2}$
$\displaystyle=\int_{B^{h}\cap B}(\phi\bm{D})(\bm{f}-\bm{f}^{h})dV^{h},$
$\displaystyle\bm{I}^{h}_{3}$ $\displaystyle=\int_{B^{h}\cap
B}(\phi\bm{D}-\phi\bm{D}^{h})\bm{f}dV^{h}.$

Consider the first term $\bm{I}^{h}_{1}$. Since $\phi\in C^{\infty}_{0}(B)$ we
have

$\mbox{supp}(\phi\bm{D}^{h})\subset B$

for any sufficiently small $h$. Hence

$\bm{I}^{h}_{1}=\int_{B^{h}}(\phi\bm{D})\bm{f}dV^{h}.$

Applying the discrete divergence theorem (Theorem 1), we get

$\displaystyle\bm{I}^{h}_{1}=\int_{\partial
B^{h}}\frac{1}{2}\sum\limits_{i=0}^{7}(\phi(x+he_{l})n_{l}^{+}(x)+\phi(x-he_{l})n_{l}^{-}(x))\bm{e}_{l}\bm{f}^{h}(x)dS^{h}(x)-\int_{B^{h}}\phi(\bm{D}^{h}f^{h})dV^{h}.$

Take $U=\mbox{supp}\phi+B(0,\frac{d}{2})$ with

$d=\min_{x\in\mbox{supp}\phi}\min_{y\in\partial B}\|x-y\|.$

Since $B^{h}$ converges to $B$, Lemma 11 indicates that

$(\mbox{supp}\phi+B(0,4h))\cap\mathbb{Z}^{8}_{h}\subset
U\cap\mathbb{Z}^{8}_{h}\subset B^{h}$

for sufficient ly small $h$. Consequently, for any

$\alpha\in(\mbox{supp}\phi+B(0,2h))\cap\mathbb{Z}^{8}_{h},$

we have

$N(\alpha)\subset B(\alpha,2h)\cap\mathbb{Z}^{8}_{h}\subset B^{h}$

so that

$(\mbox{supp}\phi+B(0,2h))\cap\mathbb{Z}^{8}_{h}\subset(B^{h})^{\circ}.$

This implies

$(\mbox{supp}\phi+B(0,2h))\cap\partial B^{h}=\emptyset$

and yields the fact that

$\int_{\partial
B^{h}}\sum\limits_{l=0}^{7}\left(\phi(\cdot+he_{l})n_{l}^{+}+\phi(\cdot-
he_{l})n_{l}^{-}\right)\bm{e}_{l}\bm{f}^{h}dS^{h}=0$

so that

$\bm{I}^{h}_{1}=-\int_{B^{h}}\phi(\bm{D^{h}f^{h}})dV^{h}=\int_{(B^{h})^{\circ}}\phi(\bm{D^{h}f^{h}})dV^{h}.$

Since $\bm{D^{h}f^{h}}=0$ on $(B^{h})^{\circ},$ $\bm{I^{h}_{1}}$ thus vanishes
for any sufficiently small $h$.

Consider the second item $\bm{I^{h}_{2}}$ in (7.8). By definition,

$\displaystyle\|\bm{I}^{h}_{2}\|$ $\displaystyle\leqslant\int_{B^{h}\cap
B}\|(\phi\bm{D}^{h})(\bm{f}-\bm{f}^{h})\|dV^{h}$
$\displaystyle\leqslant\max_{B^{h}\cap
B}\|\bm{f}-\bm{f}^{h}\|\max_{\mathbb{R}^{8}}\|\phi\bm{D}^{h}\|\int_{B^{h}\cap
B}1dV^{h}.$

Pick $R>0$ such that $B\subset B(0,R).$ We thus have

$B^{h}\cap B\subset B(0,R)\cap\mathbb{Z}^{8}_{h}.$

It is clear that

$\lim_{h\mapsto
0+}\int_{B(0,R)\cap\mathbb{Z}^{8}_{h}}1dV^{h}=\int_{B(0,R)}1dV,$

which yields

$\int_{B^{h}\cap B}1dV^{h}=O(1).$

Since $\phi\in C^{\infty}_{0}(B),$ we have

$\max_{\mathbb{R}^{8}}\|\phi\bm{D}^{h}\|=O(1).$

Hence from the assumption

$\max_{B^{h}\cap B}\|\bm{f}-\bm{f}^{h}\|=o(1),$

we derive that

$\|\bm{I}^{h}_{2}\|\leqslant o(1)O(1)O(1)=o(1).$

Now it remains to show that $\bm{I}^{h}_{3}=o(1)$. Similar to
$\bm{I}^{h}_{2},$ we have

$\displaystyle\|\bm{I}^{h}_{3}\|$ $\displaystyle\leqslant\int_{B^{h}\cap
B}\|(\phi\bm{D}-\phi\bm{D}^{h})\bm{f}\|dV^{h}$
$\displaystyle\leqslant\max_{\mbox{supp}\phi+\overline{B(0,h)}}\|\bm{f}\|\max_{\mathbb{R}^{8}}\|\phi\bm{D}-\phi\bm{D}^{h}\|\int_{B^{h}\cap
B}1dV^{h}.$

Since $\bm{f}\in C^{1}(B)$ and $\phi\in C^{\infty}_{0}(B),$ we have

(7.9)
$\max_{\mbox{supp}\phi+\overline{B(0,h)}}\|\bm{f}\|=O(1),\quad\max_{\mathbb{R}^{8}}\|\phi\bm{D}-\phi\bm{D}^{h}\|=o(1).$

As we have observed,

$\int_{B^{h}\cap B}1dV^{h}=O(1),$

we thus obtain

$\bm{I}^{h}_{3}=o(1).$

As a result,

$\bm{I^{h}_{1}+I^{h}_{2}+I^{h}_{3}}=o(1).$

This completes the proof.

∎

In conclusion, a function is deemed regular precisely when it stands as the
scaling limit of discrete regular functions.

######  Theorem 16.

Suppose $B$ is a bounded domain with $C^{3}$-boundary in $\mathbb{R}^{8}$ and
$B=B^{h}+O(h^{2})$ with $B^{h}\subset\mathbb{Z}^{8}_{h}.$ Then a function
$\bm{f}\in C^{3}(B)$ is regular if and only if there exist discrete regular
functions $\bm{f}^{h}:B^{h}\longrightarrow\mathbb{O}$ such that $\bm{f}$ is
the scaling limit of $\bm{f}^{h}$.

###### Proof.

The result is a direct consequence of merging Theorems 14 and 15. ∎

Declarations:

Funding and/or Conflicts of interests/Competing interests: This work was
supported by the NNSF of China (12171448). The authors declare that they have
no conflicts of interests and no competing interests. The authors read and
approved the final manuscript.

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