1 Introduction

Complex analytic tools together with their related discretisation methods provide a very powerful toolkit to compute numerically and some cases even analytically the solutions to many partial differential equations with given boundary data that arise in the context of harmonic analysis. In the focus there are null-solutions to a discrete version of the Cauchy–Riemann operator that satisfy a discretised version of the Cauchy integral and the Borel–Pompeiu formula. These integral formulas provide us with the key ingredients in the representation of the solutions.

Now, there are several possibilities to generalise complex function theory together with discretisations to higher dimensional settings. Many branches of engineering particularly focus on three-dimensional generalisations. To address the three-dimensional setting the function theory related to associative Clifford algebras has proven to be a very effective tool. In this context one considers a three-dimensional Cauchy–Riemann operator defined by \(D:= \sum \nolimits _{i=0}^2 \partial _{x_i} {\textbf{e}}_i\) where \({\textbf{e}}_1\) and \({\textbf{e}}_2\) are two different imaginary units generating the Clifford algebra \({\mathcal {C}}\ell _{0,2}\) which is spanned as an \({\mathbb {R}}\)-vector space by the basis elements \(1,{\textbf{e}}_1,{\textbf{e}}_2,{\textbf{e}}_1 {\textbf{e}}_2\). Actually the particular case \({\mathcal {C}}\ell _{0,2}\) represents the Hamiltonian quaternions which for a non-commutative skew-field.

Using the multiplicative structure of the Clifford algebra, the generalised first order Cauchy–Riemann operator factorises the second-order Laplacian in \({\mathbb {R}}^3\). In fact, this approach could be generalised very easily to all associative Clifford algebras. Its function theory now is broadly known under the term Clifford analysis while functions in the kernel of the generalised Cauchy–Riemann operator are often called monogenic, hyperholomorphic or Clifford holomorphic functions. Classical references are for instance the books [3, 18] as well as also J. Ryan’s edited volume containing the milestone contribution [26] among others which all provided a boost in the development of this function theory including applications to Calderon–Zygmund type operators. Particularly, [18] exhibits how related integral operators, in particular Plemelj-Sokhotzkij type formulas related to the Cauchy transform, can successfully be applied to treat boundary value problems. Additionally, J. Ryan and his co-authors have also addressed unbounded domains with this function theoretical toolkit in [19].

Over the last two decades there has been a growing interest in the discretisation of the continuous Clifford analysis for developing numerical algorithms for the higher dimensional associative framework, cf. for instance [2, 5, 6, 8, 9, 13, 14, 17] among others. However, instead of embedding the vector space \({\mathbb {R}}^{n+1}\) into associative Clifford algebras \({\mathcal {C}}\ell _{0,n}\), which have zero-divisors for \(n>2\), there are many more ways to generalise complex analysis to higher dimensions. If the Cayley–Dickson duplication process to the complex numbers is applied, then we first arrive at the four-dimensional Hamiltonian quaternions, which, as mentioned is still a particular case of a Clifford algebra. However, if we apply, as the next step, the Cayley–Dickson doubling to the quaternions, then we obtain a different sort of algebra, namely the octonions or Cayley numbers, denoted by \({\mathbb {O}}\) in all that follows, cf. for instance [1] among others. The octonions turn out to be non-associative. Consequently they do not belong to the Clifford algebras and they are not representable with matrices. But, in contrast to the eight-dimensional Clifford algebra \({\mathcal {C}}\ell _{0,3}\) octonions still form a normed division algebra (in the wider sense of being non-associative without zero-divisors). So, every non-zero element has a multiplicative inverse. According to the famous theorem of Hurwitz, they form the largest real normed zero divisor-free algebra over \({\mathbb {R}}\).

Surprisingly, although the octonions are not associative, according to recently published research results in particle physics, see for instance [4, 16, 27], they seem to offer a much more adequate model for a unified description of particle physics including gravity, which was also already proposed in the book by F. Gürsey and H. Tze in 1996, see [20]. This provides one further motivation to develop analytic and discretised function theoretic tools in the non-associative context of octonions. On the continuous level at least the fundaments of an octonionic function theory are already well-developed, see for example [12, 28, 32, 33]. In [34] generalisations of the Cauchy transform together with Plemelj projection formulas and with some basic applications to Calderon–Zygmund type operators have been presented particularly. Recently one also managed to introduce meaningful octonionic generalisations of Bergman and Hardy spaces on the continuous level, see for example [10, 11, 30, 31], as well as the papers [15, 21] addressing octonionic Hilbert spaces on a more general level.

Nevertheless, if we want to solve practical octonionic boundary value problems numerically, then we are in need of discretised versions of these octonionic operators. As far as we know, the development of a discrete octonionic function theory is still a rather open research field. In our recent paper [24], we developed a fundament for research in this direction, namely, we introduced a discretised version of octonionic Cauchy–Riemann operator in terms of appropriate forward and backward operators in the Hermitian sense, and established a discrete Stokes’ formula for that operator.

In this paper, we depart from the discretised Stokes’ formula and exploit that one further in order to obtain a discrete octonionic version of Borel–Pompeiu’s formula. The latter in turn produces a discrete Cauchy formula in a special case. As a next step, we establish a discretised version of the octonionic Cauchy transform that has been introduced on the continuous level in [34]. The related Plemelj-Sokhotzkij projection formulae then give rise to the definition of a discrete octonionic monogenic generalisation of Hardy spaces. In particular, the focus of this paper is put on the consideration of the upper octonionic half-space (resp. its related upper half-lattice) where the last component satisfies \(x_7>0\). In this case, we have a simple geometric description of the inner product that has to be used here. An alternative way to introduce discrete octonionic monogenic Hardy spaces consists in defining appropriate extension operators using the Fourier transform on the complexified octonions and its related Fourier symbols, which in our framework are also complexified octonions. Also this approach will be carefully developed in this paper as it has the advantage of being fully explicit.

As a consequence of the lack of associativity, the signs in the discrete octonionic Borel–Pompeiu and Cauchy differ from the results in discrete Clifford analysis. Surprisingly, the discretisation of the continuous octonionic analysis that we develop here has the interesting effect that the associator expressions that we intrinsically have in the Stokes formula of the continuous case (see [34]), disappear in our discretised constructions. This effect also shows up when considering instead of the forward and backward operator its centralised version to which we also pay a particular attention in this paper to get a more complete view on this particular phenomenon.

The discrete octonionic setting thus turns out to be really different as well from its associative version in Clifford analysis but also from continuous octonionic monogenic function theory. It exposes new intrinsic peculiarities that have to be taken really carefully into account.

2 Preliminaries and Notations

2.1 Continuous Octonionic Analysis

In this section, we briefly recall basic notions and the most fundamental results on continuous octonionic analysis. The basic context is the 8-dimensional Euclidean vector space \({\mathbb {R}}^{8}\), where the standard vectors are denoted by \({\textbf{e}}_{k}\), \(k=0,1,\ldots ,7\). A vector from \({\mathbb {R}}^8\) can be expressed as usual in terms of its real coordinates in the way \({\textbf{x}}=(x_{0}, x_{1},\ldots , x_{7})\). Next, \({\mathbb {R}}^8\)-vectors can also be described as octonions

$$\begin{aligned} x = x_{0}{\textbf{e}}_{0}+x_{1}{\textbf{e}}_{1}+x_{2}{\textbf{e}}_{2}+x_{3}{\textbf{e}}_{3}+x_{4}{\textbf{e}}_{4}+x_{5}{\textbf{e}}_{5}+x_{6}{\textbf{e}}_{6}+x_{7}{\textbf{e}}_{7}, \end{aligned}$$

where we now additionally identify \({\textbf{e}}_{4}={\textbf{e}}_{1}{\textbf{e}}_{2}\), \({\textbf{e}}_{5}={\textbf{e}}_{1}{\textbf{e}}_{3}\), \({\textbf{e}}_{6}={\textbf{e}}_{2}{\textbf{e}}_{3}\) and \({\textbf{e}}_{7}={\textbf{e}}_{4}{\textbf{e}}_{3}=({\textbf{e}}_{1}{\textbf{e}}_{2}){\textbf{e}}_{3}\). Moreover, we have \({\textbf{e}}_{i}^{2}=-1\) and \({\textbf{e}}_{0}{\textbf{e}}_{i}={\textbf{e}}_{i}{\textbf{e}}_{0}\) for all \(i=1,\ldots ,7\), and \({\textbf{e}}_{i}{\textbf{e}}_{j}=-{\textbf{e}}_{j}{\textbf{e}}_{i}\) for all mutual distinct \(i,j\in \left\{ 1,\ldots ,7\right\} \), as well as \({\textbf{e}}_{0}\) is the neutral element and, therefore, often will be omitted. This definition endows \({\mathbb {R}}^8\) additionally with a multiplicative closed structure.

Table 1 fully describes the multiplication rules for real octonions. This table clearly indicates in particular that the octonionic multiplication actually is closed but not associative, for instance we have \(({\textbf{e}}_{i}{\textbf{e}}_{j}){\textbf{e}}_{k}=-{\textbf{e}}_{i}({\textbf{e}}_{j}{\textbf{e}}_{k})\).

Table 1 Multiplication table for real octonions \({\mathbb {O}}\)
Full size table

Here, we use the same labelling of the basis elements as used in [1]. Additionally, we also need to consider the octonionic conjugate which is given by

$$\begin{aligned} {\overline{x}} = x_{0}{\textbf{e}}_{0}-x_{1}{\textbf{e}}_{1}-x_{2}{\textbf{e}}_{2}-x_{3}{\textbf{e}}_{3}-x_{4}{\textbf{e}}_{4}-x_{5}{\textbf{e}}_{5}-x_{6}{\textbf{e}}_{6}-x_{7}{\textbf{e}}_{7}. \end{aligned}$$

The Euclidean norm from \({\mathbb {R}}^8\) then is reproduced by \(x \cdot {\overline{x}}=\sum \nolimits _{j=0}^7 x_i^2 = |x|^2\), so every non-zero octonion x is invertible via \(x^{-1} = \dfrac{{\overline{x}}}{|x|^2}\).

While there are several possibilities to extend the classical function theory to octonions, we recall here the definition in the sense of the Riemann-approach, following the classical development of Dentoni and Sce [12], Nono [28], the school of Xingmin-Li and Zhong Peng, see for instance [32] and others. In this sense we recall:

Definition 1

(Octonionic monogenicity) Let \(U \subseteq {\mathbb {O}}\) be an open set. A real-differentiable function \(f:U \rightarrow {\mathbb {O}}\) is called left (right) octonionic monogenic if \({\mathcal {D}} f = 0\) (esp. \(f {\mathcal {D}} = 0\)). Here, \({\mathcal {D}}:= \dfrac{\partial }{\partial x_0} + \sum \nolimits _{i=1}^7 {\textbf{e}}_i \dfrac{\partial }{\partial x_i}\) is the octonionic first order Cauchy–Riemann operator. If f satisfies \(\overline{{\mathcal {D}}}f = 0\) (resp. \(f\overline{\mathcal {D}} = 0\)), then we call f left (right) octonionic anti-monogenic.

Octonionic analysis has two crucial differences to classical Clifford analysis:

  1. (i)

    Octonionic analysis considers functions from \({\mathbb {R}}^8\) back into \({\mathbb {R}}^8\), while Clifford analysis addresses null-solutions to the Cauchy–Riemann operator defined on the paravector space \({\mathbb {R}}\oplus {\mathbb {R}}^7\) with values in the Clifford algebra \({\mathcal {C}}\ell _7\). \({\mathcal {C}}\ell _7\) however is a real vector space being isomorphic to \({\mathbb {R}}^{128}\).

  2. (ii)

    Left(right) octonionic monogenic functions do neither form a right nor a left \({\mathbb {O}}\)-module. See for example Kauhanen and Orelma in [22, 23] for concrete examples. This fact significantly complicates the development of a consistent function theory and the theory of generalised Hilbert function spaces in octonionic settings, see also [10, 11, 21].

Additionally, the lack of associativity leads to modifications of the classical integral formulae, such as for example the Stokes’ formula [34]:

$$\begin{aligned}{} & {} \displaystyle \int \limits _{\partial G} g(x) \; (d\sigma (x) f(x)) \nonumber \\{} & {} \quad \displaystyle = \int \limits _G \Bigg ( g(x)({\mathcal {D}} f(x)) + (g(x){\mathcal {D}})f(x) - \sum \limits _{j=0}^7 [e_j, {\mathcal {D}}g_j(x),f(x)] \Bigg ) dV, \end{aligned}$$
(1)

Here, the expression \([a,b,c]:= (ab)c - a(bc)\) called the associator appears. This is an intrinsic feature, which is cancelled out in the cases of associativity. It is important to remark, that although the associator appears in most of octonionic constructions, it is nevertheless possible to introduce specific structures, where the associator would vanish. For example, it has been pointed out in [32], that considering the two functions being octonionic monogenic and Stein-Weiss conjugate harmonics, i.e. \(\dfrac{\partial g_j}{\partial x_i} = \dfrac{\partial g_i}{\partial x_j}\) for all \(0\le i<j\le 7\), the associator will vanish.

Further, a generalisation of the Cauchy’s integral formula to octonionic setting has been presented, see for instance [28, 33]:

Proposition 1

(Cauchy’s integral formula) Let \(U \subseteq {\mathbb {O}}\) be open and \(G \subseteq U\) be an 8-D compact oriented manifold with a strongly Lipschitz boundary \(\partial G\). If \(f:U \rightarrow {\mathbb {O}}\) is left octonionic monogenic, then for all \(x \in G\)

$$\begin{aligned} f(x)= \frac{3}{\pi ^4} \int \limits _{\partial G} q_\textbf{0}(y-x) \Big (d\sigma (y) f(y)\Big ). \end{aligned}$$

However, if the parenthesis were set differently, then one would obtain the different formula

$$\begin{aligned} \frac{3}{\pi ^4} \int \limits _{\partial G} \Big ( q_\textbf{0}(y-x) d\sigma (y)\Big ) f(y) = f(x) + \int \limits _G \sum \limits _{i=0}^7 \Big [q_\textbf{0}(y-x),{\mathcal {D}}f_i(y),{\textbf{e}}_i \Big ] dy_0 \ldots dy_7, \end{aligned}$$

which involves the associator again.

2.2 Discretisation of Octonionic Analysis

Let us consider the unbounded uniform lattice \(h{\mathbb {Z}}^{8}\) with the lattice constant \(h>0\), which is defined in the classical way as follows

$$\begin{aligned} h {\mathbb {Z}}^{8}:=\left\{ {\textbf{x}} \in {{\mathbb {R}}}^{8}\,|\, {\textbf{x}} = (m_{0}h, m_{1}h,\ldots , m_{7}h), m_{j} \in {\mathbb {Z}}, j=0,1,\ldots ,7\right\} . \end{aligned}$$

Next, we define the classical forward and backward differences \(\partial _{h}^{\pm j}\) as

$$\begin{aligned} \begin{array}{lcl} \partial _{h}^{+j}f(mh) &{}:= &{} h^{-1}(f(mh+{\textbf{e}}_jh)-f(mh)), \\ \partial _{h}^{-j}f(mh) &{}:= &{} h^{-1}(f(mh)-f(mh-{\textbf{e}}_jh)), \end{array} \end{aligned}$$
(2)

for discrete functions f(mh) with \(mh\in h{\mathbb {Z}}^{8}\). In the sequel, we consider functions defined on \(\varOmega _{h} \subset h{\mathbb {Z}}^{8}\) and taking values in octonions \({\mathbb {O}}\). As usual, all important properties such as the \(l^{p}\)-summability (\(1\le p<\infty \)) are defined component-wisely.

A next step is to introduce discretisations of the Cauchy–Riemann operators in octonions. In contrast to the classical discrete Clifford analysis presented for instance in [2, 14], where fundamental ideas of the Weyl calculus are used, we follow the alternative approach approach presented in [13] which uses a direct discretisation of the continuous Cauchy–Riemann (or Dirac) operators in terms of forward and backward finite difference operators. Following this approach, the non-associativity of octonionic multiplication can be respected, see [24] for details. Hence, by using the finite difference operators (2), we introduce a discrete forward Cauchy–Riemann operator \(D^{+}:l^{p}(\varOmega _{h},{\mathbb {O}})\rightarrow l^{p}(\varOmega _{h},{\mathbb {O}})\) and a discrete backward Cauchy–Riemann operators \(D^{-}:l^{p}(\varOmega _{h},{\mathbb {O}})\rightarrow l^{p}(\varOmega _{h},{\mathbb {O}})\) as follows

$$\begin{aligned} D^{+}_{h}:=\sum _{j=0}^{7} {\textbf{e}}_j\partial _{h}^{+j}, \quad D^{-}_{h}:=\sum _{j=0}^{7} {\textbf{e}}_j\partial _{h}^{-j}. \end{aligned}$$
(3)

Direct computations show that the star-Laplacian \(\varDelta _{h}\) is represented in this setting as follows:

$$\begin{aligned} \varDelta _{h} = \frac{1}{2}\left( D_{h}^{+}\overline{D_{h}^{-}}+D_{h}^{-}\overline{D_{h}^{+}}\right) \text{ with } \varDelta _{h}:=\sum _{j=0}^{7}\partial _{h}^{+j}\partial _h^{-j}, \end{aligned}$$

where \(\overline{D_{h}^{+}}\) and \(\overline{D_{h}^{-}}\) are the discrete conjugated forward and backward Cauchy–Riemann operators, respectively:

$$\begin{aligned} \overline{D_{h}^{-}}=\partial _{h}^{-0}-\sum _{j=1}^{7} {\textbf{e}}_j\partial _{h}^{-j}, \quad \overline{D_{h}^{+}}=\partial _{h}^{+0}-\sum _{j=1}^{7} {\textbf{e}}_j\partial _{h}^{+j}. \end{aligned}$$

Since we consider discrete forward and backward Cauchy–Riemann operators, it is also necessary to distinguish between discrete forward and backward monogenic functions:

Definition 2

A function \(f\in l^{p}(\varOmega _{h},{\mathbb {O}})\) is called discrete left forward monogenic if \(D_{h}^{+}f=0\) in \(\varOmega _{h}\). Respectively, a function \(f\in l^{p}(\varOmega _{h},{\mathbb {O}})\) is called discrete left backward monogenic if \(D_{h}^{-}f=0\) in \(\varOmega _{h}\).

2.3 Discrete Fundamental Solution

To construct discrete versions of the Borel–Pompeiu and Cauchy formulae, it is necessary to work with discrete fundamental solutions of the discrete Cauchy–Riemann operator and of the discrete Laplace operator. Thus, the following definition introduces a discrete fundamental solution to the discrete Cauchy–Riemann operators (3):

Definition 3

The function \(E_{h}^{+}:h{\mathbb {Z}}^{8} \rightarrow {\mathbb {O}}\) is called a discrete fundamental solution of \(D_{h}^{+}\) if it satisfies

$$\begin{aligned} D_{h}^{+}E_{h}^{+} =\delta _h = {\left\{ \begin{array}{ll} h^{-8}, &{}\quad \text{ for } mh=0,\\ 0,&{}\quad \text{ for } mh\ne 0, \end{array}\right. } \end{aligned}$$

for all grid points mh of \(h{\mathbb {Z}}^{8}\). Analogously, the function \(E_{h}^{-}:h{\mathbb {Z}}^{8} \rightarrow {\mathbb {O}}\) is called a discrete fundamental solution of \(D_{h}^{-}\) if it satisfies

$$\begin{aligned} D_{h}^{-}E_{h}^{-} =\delta _h = {\left\{ \begin{array}{ll} h^{-8}, &{}\quad \text{ for } mh=0,\\ 0,&{}\quad \text{ for } mh\ne 0, \end{array}\right. } \end{aligned}$$

for all grid points mh of \(h{\mathbb {Z}}^{8}\).

As usual, a discrete fundamental solution can be constructed by means of the discrete Fourier transform of \(u \in l^{p}\left( h{\mathbb {Z}}^8,{\mathbb {O}}\right) \), \(1\le p<+\infty \),

$$\begin{aligned} \mathbf {\xi } \mapsto {\mathcal {F}}_{h}u(\xi ) = \sum _{m\in {\mathbb {Z}}^{8}}e^{i\langle mh,\xi \rangle }u(mh)h^8, \quad \xi \in \left[ -\frac{\pi }{h},\frac{\pi }{h}\right] ^{8}, \end{aligned}$$

where \(\langle mh,\xi \rangle =h \sum \nolimits _{j=1}^{8}m_j \xi _j\). It is worth to underline, that the last expression implicitly introduces a complexified octonionic structure in the sense of \({\mathbb {O}}_{{\mathbb {C}}}:= {\mathbb {O}}\otimes _{{\mathbb {R}}}{\mathbb {C}}\), implying that the complex imaginary unit i commutes with all real octonions.

The inverse transform is given by \({\mathcal {F}}_{h}^{-1}={\mathcal {R}}_{h}{\mathcal {F}},\) where \({\mathcal {F}}\) is the classical continuous Fourier transform

$$\begin{aligned} x \mapsto {\mathcal {F}}f(x) = \frac{1}{(2\pi )^{8}}\int _{{\mathbb {R}}^{8}} e^{-i \langle x,\xi \rangle } f(\xi )d\xi , \end{aligned}$$

applied to an octonionic-valued function \(f \in l^{p}\left( h{\mathbb {Z}}^8,{\mathbb {O}}\right) \) with \(\textrm{supp}\, f \in \left[ -\frac{\pi }{h},\frac{\pi }{h}\right] ^{8}\), and \({\mathcal {R}}_{h}\) denotes its restriction to the lattice \(h{\mathbb {Z}}^{8}\). Note again that f is \({\mathbb {O}}\)-valued, so the expression \(e^{-i \langle x,\xi \rangle } f(\xi )\) is formally embedded in \({\mathbb {O}}_{{\mathbb {C}}}\).

Next, we recall the known symbols for the forward and backward differences \(\partial _{h}^{\pm j}\), namely \(\xi _{h}^{\pm j}=\mp h^{-1}\left( 1-e^{\mp ih\xi _j}\right) \), as well as the symbol for the star-Laplacian:

$$\begin{aligned} {\mathcal {F}}_h (\varDelta _h u )(\xi )=d^2{\mathcal {F}}_h u(\xi ) \text{ with } d^2=\frac{4}{h^2}\sum \limits _{j=0}^{7} \sin ^2\left( \frac{\xi _j h}{2}\right) . \end{aligned}$$

Thus, applying the discrete Fourier transform to forward and backward Cauchy–Riemann operators leads to

$$\begin{aligned} {\mathcal {F}}_h (D_{h}^{+} u)(\xi )=\left( \sum \limits _{j=0}^{7} {\textbf{e}}_{j}\xi _{h}^{+j}\right) {\mathcal {F}}_h u(\xi ), \quad {\mathcal {F}}_h (D_{h}^{-} u)(\xi )=\left( \sum \limits _{j=0}^{7} {\textbf{e}}_{j}\xi _{h}^{-j}\right) {\mathcal {F}}_h u(\xi ), \end{aligned}$$

implying that the operators \(D_{h}^{\pm }\) have complexified octonionic symbols \({\widetilde{\xi }}_{\pm } = \sum \nolimits _{j=0}^{7}{\textbf{e}}_{j} \xi _{h}^{\pm j}\), respectively. Hence, the fundamental solutions \(E_{h}^{\pm }\) can be expressed by

$$\begin{aligned} E_{h}^{\pm }={\mathcal {R}}_{h}{\mathcal {F}} \left( \frac{{\widetilde{\xi }}_{\pm }}{d^2}\right) =\sum \nolimits _{j=0}^{7}{\textbf{e}}_{j}{\mathcal {R}}_{h} {\mathcal {F}} \left( \frac{\xi _{h}^{\pm j}}{d^2} \right) . \end{aligned}$$
(4)

Evidently, the discrete fundamental solutions \(E_{h}^{\pm }\) constructed above are different to the discrete fundamental solutions typically considered in the framework of discrete Clifford analysis, see for example [6]. The difference comes from the fact, that we work with a direct discretisation of the Cauchy–Riemann operators without involving the splitting of basis unit vectors, as it is done in discrete Clifford analysis.

Next, we will provide two basic properties of the discrete fundamental solutions \(E_{h}^{\pm }\). For proving one of these properties, we need to recall before the following theorem [29]:

Theorem 1

Let n be the dimension of the Euclidean space and assume that \(p_{1}\), \(p_{2}\) are two positive integers with \(p_{2}<p_{1}+n\). For a positive integer \(N>0\) let \(\kappa _{N}\) consider the set of functions of the form \(T(\varTheta )=\dfrac{T_{1}(\varTheta )}{T_{2}(\varTheta )}\), \(0\ne \varTheta \in Q_{\pi }\), where \(T_{j}(\varTheta )\) are trigonometric polynomials

$$\begin{aligned} T_{j}(\varTheta ) = \sum \limits _{\mu } t_{j,\mu } e^{i \mu \cdot \varTheta }, \quad j=1,2, \end{aligned}$$

which satisfy the following conditions:

  1. (i)

    there are ordinary homogeneous polynomials \(P_{j}(\varTheta )\) of degree \(p_{j}\), \(j=1,2\), such that \(T_{j}(\varTheta )=P_{j}(\varTheta )+o(|\varTheta |^{p_{j}})\) when \(\varTheta \rightarrow 0\),

  2. (ii)

    \(|T_{2}(\varTheta )|\ge N^{-1} |\varTheta |^{p_{2}}\), \(\varTheta \in Q_{\pi }\),

  3. (iii)

    \(|t_{j,\mu }|\le N\),

  4. (iv)

    \(t_{j,\mu }=0\) for \(|\mu |>N\).

For any \(N>0\) satisfying (ii)–(iv) there is a constant C such that for all \(\mu \) (with integer components) and \(T\in \kappa _{N}\),

$$\begin{aligned} \left| ~ \int \limits _{Q_{\pi }} T(\varTheta ) e^{i \mu \cdot \varTheta } d\varTheta \right| \le C(|\mu |+1)^{-(n+p_{1}-p_{2})}. \end{aligned}$$
(5)

These tools in hand now allow us to establish the following theorem:

Theorem 2

The discrete fundamental solutions \(E_{h}^{\pm }\) to the discrete forward and backward Cauchy–Riemann operators satisfy:

  1. (i)

    \(D_{h}^{\pm }E_{h}^{\pm }(mh) = \delta _{h}(mh)\), \(m\in {\mathbb {Z}}^{8}\);

  2. (ii)

    \(E_{h}^{\pm }\in l^{p}(h{\mathbb {Z}}^{8},{\mathbb {O}})\) for \(p>\frac{8}{7}\).

Proof

The proof of property (i) can be done by a straightforward calculations. To prove (ii), we are going to use the integral representation of the discrete fundamental solution \(E_{h}^{+}\):

$$\begin{aligned} E_{h}^{+}(mh)=\frac{1}{(2\pi )^{8}}\int \limits _{\xi \in [-\frac{\pi }{h},\frac{\pi }{h}]^{8}} \frac{\tilde{\xi }_{+}}{d^{2}}e^{-i\langle mh,\xi \rangle }d\xi , \quad m\in {\mathbb {Z}}^{8}. \end{aligned}$$

Taking into account the definition of \(\tilde{\xi }_{+}\) and that

$$\begin{aligned} \xi _{h}^{+j}=-\frac{1}{h}\left( 1-e^{-ih\xi _j}\right) = -\frac{1}{h}\left( 1-\cos (h\xi _j) + i\sin (h\xi _j)\right) , \end{aligned}$$

and after using known trigonometric identity, we get

$$\begin{aligned} \begin{array}{rcl} \displaystyle \left| E_{h}^{+}(mh)\right| &{} = &{} \displaystyle \left| \frac{1}{(2\pi )^{8}} \int \limits _{\xi \in [-\frac{\pi }{h},\frac{\pi }{h}]^{8}} \frac{\xi _{h}^{+j}}{d^{2}}e^{-i\langle mh,\xi \rangle }d\xi \right| \\ \\ &{} = &{} \displaystyle \left| \frac{1}{(2\pi )^{8}} \int \limits _{\xi \in [-\frac{\pi }{h},\frac{\pi }{h}]^{8}} \frac{2\sin ^{2}\frac{h\xi _{j}}{2}}{hd^{2}}e^{-i\langle mh,\xi \rangle }d\xi \right. \\ \\ &{} &{} \displaystyle \left. + i \frac{1}{(2\pi )^{8}} \int \limits _{\xi \in [-\frac{\pi }{h},\frac{\pi }{h}]^{8}} \frac{\sin (h\xi _j)}{hd^{2}}e^{-i\langle mh,\xi \rangle }d\xi \right| . \end{array} \end{aligned}$$

The conditons of Theorem 1 are satisfied for the two integrals above if \(p_{1}=p_{2}=2\) for the first integral, and if \(p_{1}=1\), \(p_{2}=2\) for the second integral, see also [25] for a detailed discussion. Thus, the following estimate is obtained:

$$\begin{aligned} \left| \frac{1}{(2\pi )^{8}} \int \limits _{\xi \in [-\frac{\pi }{h},\frac{\pi }{h}]^{8}} \frac{\xi _{h}^{+j}}{d^{2}}e^{-i\langle mh,\xi \rangle }d\xi \right| \le \frac{C h}{\left( |mh|+h\right) ^{8}} + \frac{C}{\left( |mh|+h\right) ^{7}} \approx {\mathcal {O}}\left( \frac{1}{|mh|^{7}}\right) . \end{aligned}$$

Hence, it can be concluded that the fundamental solution \(E_{h}^{+}\) belongs to \(l^{p}(h{\mathbb {Z}}^{8},{\mathbb {O}})\) for \(p>\frac{8}{7}\). The proof for the formula of \(E_{h}^{-}\) can be performed analogously. \(\square \)

3 Discrete Stokes’ and Borel–Pompeiu Formulae

We start this section by recalling the discrete octonionic Stokes’ formulae from [24]. Again, for shortening the notations, the long list of indices \(m_{0},m_{1},\ldots ,m_{7}\), will be omitted from the argument, i.e. we will simply write f(mh) instead of \(f(m_{0}h,m_{1}h,\ldots ,m_{7}h)\).

We are particularly interested in studying upper and lower half-spaces (or half-lattices), which are defined as follows:

$$\begin{aligned} \begin{array}{rcl} h{\mathbb {Z}}_{+}^{8} &{}:= &{} \left\{ (h{\underline{m}},hm_{7}):{\underline{m}}\in {\mathbb {Z}}^{7},m_{7}\in {\mathbb {Z}}_{+}\right\} , \\ h{\mathbb {Z}}_{-}^{8} &{}:= &{} \left\{ (h{\underline{m}},hm_{7}):{\underline{m}}\in {\mathbb {Z}}^{7},m_{7}\in {\mathbb {Z}}_{-}\right\} . \end{array} \end{aligned}$$

The following two theorems express the discrete octonionic Stokes’ formulae for the upper half-lattice and for the lower half-lattice, respectively [24]:

Theorem 3

The discrete Stokes’ formula for the upper half-lattice \(h{\mathbb {Z}}_{+}^{8}\) is given by

$$\begin{aligned} \begin{array}{c} \displaystyle \sum _{m\in {\mathbb {Z}}_{+}^{8}} \left\{ \left[ g(mh)D_h^{+}\right] f(mh) - g(mh) \left[ D_h^{-}f(mh) \right] \right\} h^8 \\ \displaystyle = \sum \limits _{{\underline{m}}\in {\mathbb {Z}}^{7}} {\textbf{e}}_{7}\left( g({\underline{m}},1)f_{k}({\underline{m}},0)\right) h^{8} \end{array} \end{aligned}$$
(6)

for all discrete functions f and g such that the series converge.

Theorem 4

The discrete Stokes’ formula for the lower half-lattice \(h{\mathbb {Z}}_{-}^{8}\) is given by

$$\begin{aligned} \begin{array}{c} \displaystyle \sum _{m\in {\mathbb {Z}}_{-}^{8}} \left\{ \left[ g(mh)D_h^{+}\right] f(mh) - g(mh) \left[ D_h^{-}f(mh) \right] \right\} h^8 \\ \displaystyle = -\sum \limits _{{\underline{m}}\in {\mathbb {Z}}^{7}} {\textbf{e}}_{7}\left( g({\underline{m}},0)f_{k}({\underline{m}},-1)\right) h^{8} \end{array} \end{aligned}$$
(7)

for all discrete functions f and g such that the series converge.

Remark 1

We would like to remark that the discrete Stokes’ formulae introduced above do not contain the associator, which appears in the continuous case as an intrinsic term. The non-appearance of the associator therefore represents a surprising effect of the discrete setting. Moreover, the cancellation of the associator is not related to the consideration of the forward or backward discrete Cauchy–Riemann operators, either. Note that if we consider instead for example the central discrete Cauchy–Riemann operator:

$$\begin{aligned} {\tilde{D}}_{h}:= \frac{1}{2}\left( D_{h}^{+}+D_{h}^{-}\right) , \end{aligned}$$

then the discrete octonionic formula for the whole space would have the form

$$\begin{aligned} \begin{array}{rl}&\displaystyle \sum _{m\in {\mathbb {Z}}^{8}} \left\{ \left[ g(mh){\tilde{D}}_{h}\right] f(mh) - g(mh) \left[ {\tilde{D}}_{h}f(mh) \right] \right\} h^8 = 0. \end{array} \end{aligned}$$

For the sake of completeness, let us briefly outline the proof of this formula. We start with the first summand (h is omitted for the sake of abbreviation)

$$\begin{aligned} \begin{array}{ll} \displaystyle \sum \limits _{m\in {\mathbb {Z}}^{8}} \left[ g(m){\tilde{D}}_{h}\right] f(m)h^{8} &{}= \sum \limits _{m\in {\mathbb {Z}}^{8}} \sum \limits _{j=0}^{7} \frac{1}{2}\left[ \partial _{h}^{+j}g(m){\textbf{e}}_{j}+\partial _{h}^{-j}g(m){\textbf{e}}_{j}\right] f(m) h^{8} \\ \\ &{} = \displaystyle \sum \limits _{m\in {\mathbb {Z}}^{8}} \sum \limits _{j=0}^{7} \frac{1}{2} \sum \limits _{i=0}^{7}\sum \limits _{k=0}^{7} \left[ \partial _{h}^{+j}g_{i}(m){\textbf{e}}_{i}{\textbf{e}}_{j}+\partial _{h}^{-j}g_{i}(m){\textbf{e}}_{i}{\textbf{e}}_{j}\right] f_{k}(m){\textbf{e}}_{k} h^{8}. \end{array} \end{aligned}$$

By using the relation \(({\textbf{e}}_{i}{\textbf{e}}_{j}){\textbf{e}}_{k}=-{\textbf{e}}_{i}({\textbf{e}}_{j}{\textbf{e}}_{k})\) and the definitions of finite differences, we get

$$\begin{aligned} \begin{array}{ll} &{} \displaystyle \sum \limits _{m\in {\mathbb {Z}}^{8}} \sum \limits _{j=0}^{7}\frac{1}{2} \sum \limits _{i=0}^{7}\sum \limits _{k=0}^{7} \left[ -\left( g_{i}(m+{\textbf{e}}_{j})-g_{i}(m)\right) f_{k}(m){\textbf{e}}_{i}({\textbf{e}}_{j}{\textbf{e}}_{k})\right. \\ &{}\quad \left. \displaystyle - \left( g_{i}(m)-g_{i}(m-{\textbf{e}}_{j})\right) f_{k}(m){\textbf{e}}_{i}({\textbf{e}}_{j}{\textbf{e}}_{k})\right] h^{8} \\ \\ &{}\quad = \displaystyle \sum \limits _{m\in {\mathbb {Z}}^{8}} \sum \limits _{j=0}^{7}\frac{1}{2} \sum \limits _{i=0}^{7}\sum \limits _{k=0}^{7} \left[ -g_{i}(m+{\textbf{e}}_{j}){\textbf{e}}_{i}f_{k}(m)+g_{i}(m-{\textbf{e}}_{j}){\textbf{e}}_{i}f_{k}(m)\right] ({\textbf{e}}_{j}{\textbf{e}}_{k})h^{8}. \end{array} \end{aligned}$$

Performing a change of variables in the latter expression, we get

$$\begin{aligned} \begin{array}{ll} &{} \displaystyle \sum \limits _{m\in {\mathbb {Z}}^{8}} \sum \limits _{j=0}^{7}\frac{1}{2} \sum \limits _{i=0}^{7}\sum \limits _{k=0}^{7} \left[ -g_{i}(m){\textbf{e}}_{i}f_{k}(m-{\textbf{e}}_{j})+g_{i}(m){\textbf{e}}_{i}f_{k}(m+{\textbf{e}}_{j})\right] ({\textbf{e}}_{j}{\textbf{e}}_{k})h^{8} \\ \\ &{}\quad = \displaystyle \sum \limits _{m\in {\mathbb {Z}}^{8}} \sum \limits _{j=0}^{7} \frac{1}{2}\sum \limits _{i=0}^{7}\sum \limits _{k=0}^{7} \left[ g_{i}(m){\textbf{e}}_{i}\left( f_{k}(m+{\textbf{e}}_{j})+f_{k}(m-{\textbf{e}}_{j})\right) \right] ({\textbf{e}}_{j}{\textbf{e}}_{k})h^{8} \\ \\ &{}\quad = \displaystyle \sum \limits _{m\in {\mathbb {Z}}^{8}} \sum \limits _{j=0}^{7} \frac{1}{2}\sum \limits _{i=0}^{7}\sum \limits _{k=0}^{7} g_{i}(m){\textbf{e}}_{i}\left( \partial _{h}^{+j}{\textbf{e}}_{j} f_{k}{\textbf{e}}_{k}+\partial _{h}^{-j}{\textbf{e}}_{j} f_{k}{\textbf{e}}_{k}\right) h^{8} \\ \\ &{}\quad = \displaystyle \sum \limits _{m\in {\mathbb {Z}}^{8}} g(m)\left[ {\tilde{D}}_{h}f(m)\right] h^{8}. \end{array} \end{aligned}$$

We just have proved the discrete octonionic Stokes’ formula for the whole space in the case of the central discrete Cauchy–Riemann operator. This highlights that the constructions in the discrete settings exhibit an essentially different nature than the constructions in the continuous case, where the associstor appears as an intrinsic ingredient.

By means of the discrete discrete octonionic Stokes’ formule (6)-(7), the discrete octononionic Borel–Pompeiu formulae can be introduced:

Theorem 5

Let \(E_{h}^{+}\) be the discrete fundamental solution to the discrete Cauchy–Riemann operator \(D_{h}^{+}\). Then the discrete octonionic Borel–Pompeiu formula for the upper half-lattice \(h{\mathbb {Z}}_{+}^{8}\) is given by

$$\begin{aligned} \begin{array}{l} \displaystyle \sum _{n\in {\mathbb {Z}}_{+}^{8}} E_{h}^{+}(nh-mh) \left[ D_h^{-}f(mh) \right] h^8 \\ \displaystyle \quad + \sum \limits _{{\underline{n}}\in {\mathbb {Z}}^{7}} {\textbf{e}}_{7}\left( E_{h}^{+}({\underline{n}}h-{\underline{m}}h,1)f_{k}({\underline{m}},0)\right) h^{8} = \left\{ \begin{array}{ll} 0, &{}\quad m\notin {\mathbb {Z}}_{+}^{8}, \\ f(mh), &{}\quad m\in {\mathbb {Z}}_{+}^{8}, \end{array} \right. \end{array} \end{aligned}$$
(8)

for any discrete function f such that the series converge.

Proof

To prove the discrete octonionic Borel–Pompeiu formula for the upper half-lattice, we use at first the discrete octonionic Stokes’s formula for the upper half-lattice (6), and replace therein the function g by the shifted discrete fundamental solution \(E_{h}^{+}(\cdot -mh)\) with \(m\in {\mathbb {Z}}_{+}^{8}\). Next, by taking into account the definition of the discrete fundamental solution, we note that \(\left[ E_{h}^{+}(nh-mh)D_{h}^{+}\right] =0\) for \(n \ne m\) and \(\left[ E_{h}^{+}(nh-mh)D_{h}^{+}\right] =h^{-8}\) for \(n=m\). Thus, the formula is established. \(\square \)

By using the discrete octonionic Borel–Pompeiu formula (8) and requiring that the function f is discrete left backward monogenic in \(hZ_{+}^{8}\), we immediately arrive at the discrete octonionic Cauchy formula:

Theorem 6

Let f be a discrete left backward monogenic function with respect to the operator \(D_{h}^{-}\), and let \(E_{h}^{+}\) be the discrete fundamental solution to the operator \(D_{h}^{+}\). Then the discrete octonionic Cauchy formula for the upper half-lattice \(hZ_{+}^{8}\) is given by

$$\begin{aligned} \sum \limits _{{\underline{n}}\in {\mathbb {Z}}^{7}} {\textbf{e}}_{7}\left( E_{h}^{+}({\underline{n}}h-{\underline{m}}h,1)f_{k}({\underline{m}},0)\right) h^{8} = \left\{ \begin{array}{ll} 0, &{}\quad m\notin {\mathbb {Z}}_{+}^{8}, \\ f(mh), &{}\quad m\in {\mathbb {Z}}_{+}^{8}, \end{array} \right. \end{aligned}$$
(9)

which holds for any discrete function f such that the series converge.

Analogously we may introduce discrete octonionic Borel–Pompeiu and Cauchy formulae for the lower half-lattice \(h{\mathbb {Z}}_{+}^{8}\):

Corollary 1

Let f be a discrete left backward monogenic function with respect to the operator \(D_{h}^{-}\), and let \(E_{h}^{+}\) be the discrete fundamental solution to operator \(D_{h}^{+}\). Then the discrete octonionic Borel–Pompeiu formula for the lower half-lattice \(h{\mathbb {Z}}_{-}^{8}\) is given by

$$\begin{aligned} \begin{array}{l} \displaystyle \sum _{n\in {\mathbb {Z}}_{-}^{8}} E_{h}^{+}(nh-mh) \left[ D_h^{-}f(mh) \right] h^8 \\ \displaystyle \quad - \sum \limits _{{\underline{n}}\in {\mathbb {Z}}^{7}} {\textbf{e}}_{7}\left( E_{h}^{+}({\underline{n}}h-{\underline{m}}h,0)f_{k}({\underline{m}},-1)\right) h^{8} = \left\{ \begin{array}{ll} 0, &{}\quad m\notin {\mathbb {Z}}_{-}^{8}, \\ f(mh), &{}\quad m\in {\mathbb {Z}}_{-}^{8}, \end{array} \right. \end{array} \end{aligned}$$
(10)

for any discrete function f such that the series converge. In the case when f is a discrete left backward monogenic function with respect to the operator \(D_{h}^{-}\), we obtain the discrete octonionic Cauchy formula for the lower half-lattice \(h{\mathbb {Z}}_{-}^{8}\) in the form

$$\begin{aligned} -\sum \limits _{{\underline{n}}\in {\mathbb {Z}}^{7}} {\textbf{e}}_{7}\left( E_{h}^{+}({\underline{n}}h-{\underline{m}}h,0)f_{k}({\underline{m}},-1)\right) h^{8} = \left\{ \begin{array}{ll} 0, &{}\quad m\notin {\mathbb {Z}}_{-}^{8}, \\ f(mh), &{}\quad m\in {\mathbb {Z}}_{-}^{8}. \end{array} \right. \end{aligned}$$
(11)

Theorem 6 and Corollary 1 immediately lead to the definition of discrete octonionic Cauchy transforms for the upper and lower half-lattices:

Definition 4

For a discrete \(l^p\)-function f, \(1\le p<+\infty \), defined on the boundary layers \(({\underline{m}},0)\) and \(({\underline{m}},1)\) the discrete octonionic Cauchy transform for the upper half-lattice \(h{\mathbb {Z}}_{+}^{8}\) is defined by

$$\begin{aligned} {\mathcal {C}}_{{\mathbb {O}}}^{+}[f](mh):= \sum \limits _{{\underline{n}}\in {\mathbb {Z}}^{7}} {\textbf{e}}_{7}\left( E_{h}^{+}({\underline{n}}h-{\underline{m}}h,1)f_{k}({\underline{m}},0)\right) h^{8}. \end{aligned}$$
(12)

Analogously, for a discrete \(l^p\)-function f, \(1\le p<+\infty \), defined on the boundary layers \(({\underline{m}},0)\) and \(({\underline{m}},-1)\) the discrete octonionic Cauchy transform for the lower half-lattice \(h{\mathbb {Z}}_{-}^{8}\) is defined by

$$\begin{aligned} {\mathcal {C}}_{{\mathbb {O}}}^{-}[f](mh):= -\sum \limits _{{\underline{n}}\in {\mathbb {Z}}^{7}} {\textbf{e}}_{7}\left( E_{h}^{+}({\underline{n}}h-{\underline{m}}h,0)f_{k}({\underline{m}},-1)\right) h^{8}. \end{aligned}$$
(13)

It is worth to mention, that alike in the continuous case also in discrete Clifford analysis, the discrete octonionic Cauchy formulae and Cauchy transform indicate the dependence of (discrete) monogenic functions on their boundary values. However, as a consequence of the lack of associativity, the signs in the discrete octonionic Borel–Pompeiu and Cauchy formulae (8)–(11) differ from the results in discrete Clifford analysis, compare for example with [6]. Additionally, in line with the discussion in [24], the discretisation of the continuous octonionic analysis has the interesting effect that the associator disappears from the constructions.

Let us now present some properties of the discrete octonionic Cauchy transforms:

Theorem 7

Let us consider the discrete upper half-lattice \(h{\mathbb {Z}}_{+}^{8}\) and the lower half-lattice \(h{\mathbb {Z}}_{-}^{8}\). Then the discrete Cauchy transforms (12)–(13) satisfy the following properties:

  1. (i)

    The interior and exterior Cauchy transforms have the following mapping properties:

    $$\begin{aligned} \begin{array}{ll} \displaystyle {\mathcal {C}}_{{\mathbb {O}}}^{+} :l^{p}(h{\mathbb {Z}}_{+}^{8},{\mathbb {O}}) \rightarrow l^{q}(h{\mathbb {Z}}_{+}^{8},{\mathbb {O}}), &{} 1 \le p,q \le \infty , \\ \displaystyle {\mathcal {C}}_{{\mathbb {O}}}^{-} :l^{p}(h{\mathbb {Z}}_{-}^{8},{\mathbb {O}}) \rightarrow l^{q}(h{\mathbb {Z}}_{-}^{8},{\mathbb {O}}), &{} 1 \le p< \infty , \frac{8}{7}< q < \infty . \\ \end{array} \end{aligned}$$
  2. (ii)

    \(D_{h}^{+} {\mathcal {C}}_{{\mathbb {O}}}^{+}[f](mh)=0\), \(\forall \, m=({\underline{m}},m_{7})\) with \(m_{7}\ge 1\).

  3. (iii)

    \(D_{h}^{+} {\mathcal {C}}_{{\mathbb {O}}}^{-}[f](mh)=0\), \(\forall \, m=({\underline{m}},m_{7})\) with \(m_{7}\le -1\).

Proof

The proof of this theorem can be performed along the same ideas as in [6, 9], and, therefore, we will only mention that the first statement follows from a direct application of Hölder’s inequality and the properties of the discrete fundamental solution \(E_{h}^{+}\). The proof of (ii) and (iii) is done by straightforward application of the discrete forward Cauchy–Riemann operator \(D_{h}^{+}\) and applying its properties. \(\square \)

4 Discrete Octonionic Hardy Spaces

The aim of this section is to construct discrete octonionic Hardy spaces. Looking at the results in discrete Clifford analysis related to Hardy spaces, see for example [6, 8, 9], it becomes evident that a typical approach to introduce discrete Hardy spaces will be to work with Fourier transforms on boundary layers of the discrete fundamental solution. In this way, discrete Riesz kernels can be defined, and, hence, discrete Plemelj (or Hardy) projections are then introduced. Alternatively, one could follow the classical continuous approach for defining Hardy spaces and adapt it to the discrete setting. For the purpose of a better understanding the effect of non-associativity, both approaches will be presented in this section.

At first, we recall from [10] the following definition of an octonionic Hilbert space, which straightforwardly extends to the discrete setting:

Definition 5

An octonionic Hilbert space H is a left \({\mathbb {O}}\)-module with an octonion-valued inner product \(\langle \cdot ,\cdot \rangle :H\times H\rightarrow {\mathbb {O}}\) such that \(\left( H,\langle \cdot ,\cdot \rangle _{0}\right) \) is a real Hilbert space, where \(\langle \cdot ,\cdot \rangle _{0}:= \textrm{Re}\langle \cdot ,\cdot \rangle \). The octonion-valued inner product is supposed to satisfy for all \(f,g,h\in H\) and for all \(\alpha \in {\mathbb {O}}\) the following rules:

  1. (i)

    Additivity: \(\langle f+g,h\rangle = \langle f,h \rangle + \langle g,h \rangle \);

  2. (ii)

    Hermitian property: \(\langle g,f \rangle = \overline{\langle f,g \rangle }\);

  3. (iii)

    Strict positivity: \(\langle f,f \rangle \in {\mathbb {R}}^{\ge 0}\) and \(\langle f,f \rangle = 0\) iff \(f=0\);

  4. (iv)

    \({\mathbb {R}}\)-homogeneity: \(\langle fr,g \rangle = \langle f,g \rangle r\) for all \(r\in {\mathbb {R}}\);

  5. (v)

    \({\mathbb {O}}\)-homogeneity: \(\langle f\alpha ,f \rangle = \langle f,f \rangle \alpha \);

  6. (vi)

    \({\mathbb {O}}\)-para-linearity: \(\langle f\alpha ,g\rangle _{0} = \textrm{Re}\left( \langle f\alpha ,g\rangle \right) = \textrm{Re}\left( \langle f,g\rangle \alpha \right) \).

Before introducing the notion of a discrete octonionic Hardy space, let us fix the notation \(\gamma _{h}\) for the discrete boundary of \(\varOmega _{h}\) (0-layer in the case of half-lattices). Now we introduce the following notion of a discrete octonionic Hardy space, which is adapted from the definition presented in [10]:

Definition 6

The discrete octonionic Hardy space \({\mathcal {H}}_{h}^{2}(\varOmega _{h},{\mathbb {O}})\) is the closure of the set of \(l^{2}(\gamma _{h},{\mathbb {O}})\) functions that are discrete left monogenic inside of \(\varOmega _{h}\) and extendable to the boundary \(\gamma _{h}\).

It is important to remark, that we have not distinguished in this definition between discrete forward and backward monogenic functions as it has been introduced in Definition 2. The reason for this is to avoid too many specific definitions, which do not generally change the notion of a discrete octonionic Hardy space, since both types of discrete monogenic functions can be used. Further, Definition 6 requires a discrete monogenic function being extendable to the boundary \(\gamma _{h}\), implying that an extension operator must be defined. Although there are several possibilities to define such an operator, arguably the most straightforward way is to work with the Fourier transform of the discrete fundamental solution \(E_{h}^{+}\), as it is done in the discrete Clifford analysis, see again [6, 8, 9]. Hence, we will introduce an extension operator based on this idea, and thus, establish a clear connection between the discrete Clifford analysis approach and the “continuous theory-based”  approach.

Next, we need to introduce an octonion-valued inner product on the discrete boundary \(\gamma _{h}\):

Definition 7

For discrete octonionic function \(f,g\in l^{2}(\gamma _{h})\), we introduce the following \({\mathbb {O}}\)-valued inner products:

$$\begin{aligned} \langle f,g\rangle _{\gamma _{h}}^{h{\mathbb {Z}}_{+}^{8}}:= \sum _{mh\in \gamma _{h}} \overline{\left( -{\textbf{e}}_{7}g(mh)\right) }\left( -{\textbf{e}}_{7}f(mh)\right) h^{2}, \end{aligned}$$
(14)

if the upper half-lattice is considered, and

$$\begin{aligned} \langle f,g\rangle _{\gamma _{h}}^{h{\mathbb {Z}}_{-}^{8}}:= \sum _{mh\in \gamma _{h}} \overline{\left( {\textbf{e}}_{7}g(mh)\right) }\left( {\textbf{e}}_{7}f(mh)\right) h^{2}, \end{aligned}$$
(15)

if the lower half-lattice is considered.

It is then easy to verify the following statement:

Proposition 2

The sets \(\left( {\mathcal {H}}_{h}^{2}(\gamma _{h},{\mathbb {O}}),\langle \cdot ,\cdot \rangle _{\gamma _{h}}^{h{\mathbb {Z}}_{+}^{8}}\right) \) and \(\left( {\mathcal {H}}_{h}^{2}(\gamma _{h},{\mathbb {O}}),\langle \cdot ,\cdot \rangle _{\gamma _{h}}^{h{\mathbb {Z}}_{-}^{8}}\right) \) are octonionic Hilbert spaces in the sense of Definition 5.

The next step is to establish the theory of discrete Hardy spaces in the sense of discrete Clifford analysis. This requires a definition of discrete Riesz kernels (convolution kernels) implying that the behaviour of the discrete fundamental solution \(E_{h}^{+}\) on boundary layers needs to be studied. Let us recall the integral representation of the discrete fundamental solution

$$\begin{aligned} E_{h}^{+}(mh)=\frac{1}{(2\pi )^{8}}\int \limits _{\xi \in [-\frac{\pi }{h},\frac{\pi }{h}]^{8}} \frac{\tilde{\xi }_{+}}{d^{2}}e^{-i\langle mh,\xi \rangle }d\xi , \quad m\in {\mathbb {Z}}^{8} \end{aligned}$$

with \({\widetilde{\xi }}_{+} = \sum \nolimits _{j=0}^{7}{\textbf{e}}_{j} \xi _{h}^{+j}\). To study Fourier symbols on the boundary layers, i.e. for \(m_{7}\in \left\{ -1,0,1\right\} \), we apply the 7–dimensional discrete Fourier transform to the discrete fundamental solution:

$$\begin{aligned} \displaystyle {\mathcal {F}}_h^{(7)} E_{h}^{+}({\underline{\eta }}, m_{7}h)= & {} \sum _{{\underline{mh}} \in h{\mathbb {Z}}^{7}} e^{-i h \langle {\underline{m}}, {\underline{\eta }} \rangle } \left[ \frac{1}{(2\pi )^8} \int \limits _{\left[ -\frac{\pi }{h},\frac{\pi }{h}\right] ^{7} } e^{-i h \langle m, \xi \rangle } \frac{{\widetilde{\xi }}_{+}}{d^2} d\xi \right] \\ \displaystyle= & {} \frac{1}{(2\pi )^{7}} \int \limits _{\left[ -\frac{\pi }{h}, \frac{\pi }{h} \right] ^{7}} \sum _{{\underline{mh}} \in h{\mathbb {Z}}^{7}} e^{-i h \langle \underline{m}, {\underline{\eta }} - {\underline{\xi }} \rangle } \underbrace{\left[ \frac{1}{2\pi } \int \limits _{-\frac{\pi }{h}}^{\frac{\pi }{h}} e^{-ih m_{7}\xi _{7} } \frac{{\widetilde{\xi }}_{+}}{d^2} d\xi _{7}\right] }_{(I)} d\underline{\xi }. \end{aligned}$$

Let us study now the integral (I):

$$\begin{aligned} \begin{array}{lcl} (I) &{} = &{} \displaystyle \frac{1}{2\pi } \int \limits _{-\frac{\pi }{h}}^{\frac{\pi }{h}} e^{-ih m_{7}\xi _{7} } \frac{{\widetilde{\xi }}_{+}}{d^2} d\xi _{7} = \frac{1}{2\pi } \int \limits _{-\frac{\pi }{h}}^{\frac{\pi }{h}} e^{-ih m_{7}\xi _{7} } \frac{\widetilde{\underline{\xi }}_{+}+{\widetilde{\xi }}_{+,7}}{{\underline{d}}^2 + \frac{4}{h^{2}}\sin ^{2}\left( \frac{\xi _{7}h}{2}\right) } d\xi _{7} \\ &{} = &{} \displaystyle \frac{1}{2\pi } \int \limits _{-\frac{\pi }{h}}^{\frac{\pi }{h}} e^{-ih m_{7}\xi _{7} } \frac{\widetilde{\underline{\xi }}_{+}-{\textbf{e}}_{7}\frac{1}{h}\left( 1-e^{-ih\xi _{7}}\right) }{{\underline{d}}^2 + \frac{4}{h^{2}}\sin ^{2}\left( \frac{\xi _{7}h}{2}\right) } d\xi _{7} \\ &{} = &{} \displaystyle \frac{\widetilde{\underline{\xi }}_{+}}{2\pi } \int \limits _{-\frac{\pi }{h}}^{\frac{\pi }{h}} \frac{e^{-ih m_{7}\xi _{7} }}{{\underline{d}}^2 + \frac{4}{h^{2}}\sin ^{2}\left( \frac{\xi _{7}h}{2}\right) } d\xi _{7} - \frac{{\textbf{e}}_{7}}{2\pi h} \int \limits _{-\frac{\pi }{h}}^{\frac{\pi }{h}} e^{-ih m_{7}\xi _{7} } \frac{1-e^{-ih\xi _{7}}}{{\underline{d}}^2 + \frac{4}{h^{2}}\sin ^{2}\left( \frac{\xi _{7}h}{2}\right) } d\xi _{7} \\ &{} = &{} \displaystyle \frac{\widetilde{\underline{\xi }}_{+}}{2\pi } \int \limits _{-\frac{\pi }{h}}^{\frac{\pi }{h}} \frac{e^{-ih m_{7}\xi _{7} }}{{\underline{d}}^2 + \frac{4}{h^{2}}\sin ^{2}\left( \frac{\xi _{7}h}{2}\right) } d\xi _{7} - \frac{{\textbf{e}}_{7}}{2\pi h} \int \limits _{-\frac{\pi }{h}}^{\frac{\pi }{h}} \frac{e^{-ih m_{7}\xi _{7} }}{{\underline{d}}^2 + \frac{4}{h^{2}}\sin ^{2}\left( \frac{\xi _{7}h}{2}\right) } d\xi _{7} \\ &{} &{} \displaystyle + \frac{{\textbf{e}}_{7}}{2\pi h} \int \limits _{-\frac{\pi }{h}}^{\frac{\pi }{h}} \frac{e^{-ih\xi _{7}(m_{7}+1)}}{{\underline{d}}^2 + \frac{4}{h^{2}}\sin ^{2}\left( \frac{\xi _{7}h}{2}\right) } d\xi _{7}. \end{array} \end{aligned}$$

The integrals above need to be calculated for \(m_{7}\in \left\{ -1,0,1\right\} \), and these calculations have been presented in [6] for the case of splitting of basis unit elements. Hence, the Fourier symbols of the discrete fundamental solutions on the layers \(m_{7}=-1\), \(m_{7}=0\), and \(m_{7}=1\) are given by

$$\begin{aligned} \begin{array}{rcl} \displaystyle {\mathcal {F}}_h^{(7)} E_{h}^{+}({\underline{\xi }}, 0) &{} = &{} \displaystyle \frac{\widetilde{\underline{\xi }}_{+}}{{\underline{d}}\sqrt{4+h^{2}{\underline{d}}^{2}}} - {\textbf{e}}_{7}\left( \frac{1}{2}-\frac{h{\underline{d}}}{2\sqrt{4+h^{2}{\underline{d}}^{2}}}\right) , \\ \displaystyle {\mathcal {F}}_h^{(7)} E_{h}^{+}({\underline{\xi }}, h) &{} = &{} \displaystyle \frac{\widetilde{\underline{\xi }}_{+}}{{\underline{d}}}\left( \frac{2+h^{2}{\underline{d}}^{2}}{2\sqrt{4+h^{2}{\underline{d}}^{2}}}-\frac{h{\underline{d}}}{2}\right) + {\textbf{e}}_{7}\left( -\frac{3h{\underline{d}}+h^{3}{\underline{d}}^{3}}{2\sqrt{4+h^{2}{\underline{d}}^{2}}}+\frac{h^{2}{\underline{d}}^{2}+1}{2}\right) , \\ \displaystyle {\mathcal {F}}_h^{(7)} E_{h}^{+}({\underline{\xi }}, -h) &{} = &{} \displaystyle \frac{\widetilde{\underline{\xi }}_{+}}{{\underline{d}}}\left( \frac{2+h^{2}{\underline{d}}^{2}}{2\sqrt{4+h^{2}{\underline{d}}^{2}}}-\frac{h{\underline{d}}}{2}\right) - {\textbf{e}}_{7}\left( -\frac{3h{\underline{d}}+h^{3}{\underline{d}}^{3}}{2\sqrt{4+h^{2}{\underline{d}}^{2}}}+\frac{h^{2}{\underline{d}}^{2}+1}{2}\right) . \\ \end{array} \end{aligned}$$

Next, following ideas in [6], we introduce the pair of operators:

$$\begin{aligned} \begin{array}{rcl} H_{+}f &{}:= &{} \displaystyle {\mathcal {F}}_{h}^{-1}\left[ {\textbf{e}}_{7}\frac{\widetilde{\underline{\xi }}_{+}}{{\underline{d}}}\frac{2}{h{\underline{d}}-\sqrt{4+h^{2}{\underline{d}}^{2}}}\right] {\mathcal {F}}_{h}f, \\ H_{-}f &{}:= &{} \displaystyle -{\mathcal {F}}_{h}^{-1}\left[ {\textbf{e}}_{7}\frac{\widetilde{\underline{\xi }}_{+}}{{\underline{d}}}\frac{h{\underline{d}}-\sqrt{4+h^{2}{\underline{d}}^{2}}}{2}\right] {\mathcal {F}}_{h}f, \end{array} \end{aligned}$$

which fulfil the condition \((H_+)^2 = (H_- )^2=I\). By help of these operators, we can formulate conditions for a function to be a discrete boundary value of a discrete octonionic monogenic function in \(h{\mathbb {Z}}_{+}^{8}\) or \(h{\mathbb {Z}}_{-}^{8}\):

$$\begin{aligned} \begin{array}{lcl} f(mh) &{} = &{} H_+f(mh), \text{ for } m_{7}=1, \\ f(mh) &{} = &{} H_-f(mh), \text{ for } m_{7}=-1. \end{array} \end{aligned}$$

This enables us now to introduce another definition of discrete octonionic Hardy spaces:

Definition 8

The space of discrete functions \(f\in l^{p}(h{\mathbb {Z}}_{+}^{8},{\mathbb {O}})\) whose discrete 7D-Fourier transform fulfills \(f= H_+f\) for \(m_{7}=1\) is called the upper discrete octonionic Hardy space and it is denoted by \(h_{p,h{\mathbb {Z}}_{+}^{8}}^{+}\). Analogously, the space of discrete functions \(f\in l^{p}(h{\mathbb {Z}}_{-}^{8},{\mathbb {O}})\) whose discrete 7D-Fourier transform fulfills \(f= H_-f\) for \(m_{7}=-1\) is called the lower discrete octonionic Hardy space and it is denoted by \(h_{p,h{\mathbb {Z}}_{-}^{8}}^{-}\).

By means of the operators \(H_{+}\) and \(H_{-}\), the discrete Plemelj or Hardy projections can be now introduced as follows

$$\begin{aligned} P_+=\frac{1}{2}\left( I+ H_+ \right) \text{ and } P_-=\frac{1}{2}\left( I+H_-\right) . \end{aligned}$$

Additionally, combining these projections with the previous definition we get

$$\begin{aligned} f\in h_{p,h{\mathbb {Z}}_{+}^{8}}^{+} \Longleftrightarrow P_{+}f=f, \text{ and } f\in h_{p,h{\mathbb {Z}}_{-}^{8}}^{-} \Longleftrightarrow P_{-}f=f. \end{aligned}$$

Finally, we define two extension operators, which extend a discrete function from layers \(m_{7}=-1\) and \(m_{7}=1\) to the boundary layer \(m_{7}=0\), see [7] for the details:

Definition 9

The upper extension operator, denoted as \({\mathcal {A}}_{+}\), is an operator extending a function given on the boundary layer \(m_{7}=1\) to the boundary layer \(m_{7}=0\), i.e. it is a mapping \({\mathcal {A}}_{+} :l^{p}(h{\mathbb {Z}}^{7}) \rightarrow l^{p}(h{\mathbb {Z}}^{7})\) given by

$$\begin{aligned} {\mathcal {A}}_{+}:= {\mathcal {F}}_h^{(n-1)} \left[ \frac{ \widetilde{\underline{\xi }}_{+}}{{\underline{d}}} \left( \dfrac{2}{\sqrt{4+h^{2} {\underline{d}}^{2}} - h {\underline{d}}} \right) \right] . \end{aligned}$$

Similarly, the lower extension operator, denoted by \({\mathcal {A}}_{-}\), is an operator extending a function given on the boundary layer \(m_{7}=-1\) to the boundary layer \(m_{7}=0\), i.e. it is a mapping \({\mathcal {A}}_{-}:l^{p}(h{\mathbb {Z}}^{7}) \rightarrow l^{p}(h{\mathbb {Z}}^{7})\) given by

$$\begin{aligned} {\mathcal {A}}_{-}:= {\mathcal {F}}_h^{(n-1)} \left[ \left( \frac{\sqrt{4+h^{2}{\underline{d}}^{2}} + h{\underline{d}}}{\sqrt{4+h^{2} {\underline{d}}^{2}} - h {\underline{d}}} \right) \right] . \end{aligned}$$

This definition rounds off the discussion around discrete octonionic Hardy spaces, which we have initiated in Definition 6. Now it is clear how an extension of a discrete function to the boundary layer \(m_{7}=0\) can be explained.

5 Summary

In this paper, we have continued the development of discrete octonionic analysis by introducing discrete Borel–Pompeiu and Cauchy formulae, and defined discrete octonionic Hardy spaces for half-spaces. Moreover, we have discussed two approaches for constructing discrete Hardy spaces: (i) by a direct discretisation of the continuous case; and (ii) by using an approach used in discrete Clifford analysis by studying Fourier symbols of the discrete fundamental solution of the discrete Cauchy–Riemann operator on the boundary layers. Both approaches complement each other and contribute to a better understanding of the discrete octonionic setting.

A very surprising result is that the associator, which appears in the continuous case, does not appear in the discrete setting. For a better understanding of this effect, the discrete Stokes’ formula for the whole space has been proved also for the central discrete Cauchy–Riemann operator. Nevertheless, the associator does not appear even in that case, which underlines the particularity of the constructions in the discrete setting.

The results presented in this paper provide us with a powerful basic toolkit for a further development of discrete octonionic analysis. In particular, the discrete versions of octonionic Hardy spaces that we introduced allow us to study concrete boundary value problems for monogenic functions in the discrete octonionic setting. This also opens the door to study Calderon–Zygmund type operators in this context. Furthermore, the discrete toolkit allow us us develop numerical algorithms for solving some particular physical problems arising in the unification of particle physics and gravity as illustrated in a series of recent works in this direction, see again [4, 27], such as already mentioned roughly in the introductory text.

Further, after understanding the difficulties arising on the way discretising octonionic analysis for the half-space settings, the results of this paper can subsequentially be extended to the case of considering bounded domains in \({\mathbb {R}}^{8}\).