Draft:Polyanalyticity

Polyanalyticity.

Background and Motivation

The standard theory of polyanalytic functions is a sub-branch of complex analysis. As the name suggests, polyanalyticity and polyanalytic functions are the perhaps most natural generalizations of complex analyticity and complex analytic functions. The latter being the key objects in the field of complex analysis. In order for the reader to understand the context and to appreciate the notion of polyanalyticity, we assume the reader is familiar with complex analytic functions (also called holomorphic functions). There are different equivalent ways to define complex analytic functions. From the perspective of partial differential equations they can be identified as the set of functions annihilated by the Cauchy-Riemann operator ${\frac {\partial }{\partial {\bar {z}}}}$. Powers of the Cauchy-Riemann operator where present in relation to elasticity problems studied by Kolosov (1909), whose studies involved so called bianalytic functions which are functions of the form $a(z)+{\bar {z}}b(z)$, where $a$ and $b$ are holomorphic. Before that Goursat had studied so called biharmonic functions (constituting the kernel of the square of the Laplace operator $\Delta ^{2}$) in particular proving that any biharmonic function $u$ can be identified as the real part of what was later called bianalytic functions. Burgatti and Theodorescu where among the first to initiate a general study of the kernels of the operators $\left({\frac {\partial }{\partial {\bar {z}}}}\right)^{q}$ for positive integers $q$ (i.e. powers of the Cauchy-Riemann operator). As it turns out, many properties of complex analytic functions (the case $q=1$) have similar counterparts in the higher order situation ($q>1$) for example integral and power series representations, automatic real-analyticity of solutions to the defining equations and the structure of the sets of uniqueness. One way to characterize polyanalyticity in finite dimension is precisely by replacing the system $\partial _{{\bar {z}}_{j}}f=0,j=1,\ldots ,n,$ for functions defined on open subsets of $\mathbb {C} ^{n}$, by the system $\partial _{{\bar {z}}_{j}}^{\alpha _{j}}f=0,j=1,\ldots ,n,\alpha \in \mathbb {Z} _{+}^{n}.$ There is in finite higher dimensional complex analysis an alternative notion, due to Ahern & Bruna which may from a certain perspective, be more natural and which conforms to the possibility of representing complex analytic functions in terms of homogeneous series (which in particular renders the property that the restriction to each complex slice is again complex analytic).

A classical comprehensive survey of polyanalytic functions in one complex variable is due to Balk & Zuev (1970), and later Balk (1991). From the perspective of boundary value problems, a standard reference in the case of both one complex dimension and higher (finite) complex dimension is due to Begehr. Proofs of all propositions in this article can be found in the survey of Daghighi , which includes the most basic original result in the field and also some natural generalizations and analogues of the notion of polyanalyticity (as described in later sections).

Definition in the Case of Complex Analysis of One Variable

The following terminology was proposed by Burgatti in 1922.

Definition 1 ($q$-analytic function). Let $\Omega \subset \mathbb {C}$ be an open subset and let $q\in \mathbb {Z} _{+}$. A distribution solutions, $f$, to $\partial _{\bar {z}}^{q}f=0$ on $\Omega$ is called a $q$-analytic function on $\Omega$.

As pointed out by Balk there are many different equivalent definitions of and terminology used simultaneously for what we call $q$-analytic functions and Balk uses interchangeably $n$-analytic and polyanalytic of order $n$ with a preference for the latter.

Definition 2 (Polyanalytic function of order $q$). Let $\Omega \subseteq \mathbb {C}$ be a domain and let $q\in \mathbb {Z} _{+}$. A function $f$ is called polyanalytic of order $q$ at $p_{0}$ if it can, near $p_{0}$, be represented in the form:

$f(z)=\sum _{j=0}^{q-1}a_{j}(z){\bar {z}}^{j}$ 1

where $a_{j},j=0,\ldots ,q-1,$ are holomorphic functions near $p_{0}$. The case $a_{q-1}\equiv 0$ is not excluded. When $a_{q-1}\not \equiv 0$ the number $q$ is called the exact order of polyanalyticity of $f$. A function is called polyanalytic of order $q$ on $\Omega$ if it is polyanalytic of order $q$ at each point of $\Omega$. The space of polyanalytic functions on $\Omega$ is denoted by ${\mbox{PA}}_{q}(\Omega )$. The functions $a_{j}$ are called the analytic (or holomorphic) components of $f$.

Note that since $\Omega$ is connected, each of the local analytic components of $f(z)$ extend to a global (on $\Omega$) analytic component. Definition 1 is equivalent to Definition 2.

Proposition 1. Let $\Omega \subseteq \mathbb {C}$ be a domain and let $q\in \mathbb {Z} _{+}$. Then ${\mbox{PA}}_{q}(\Omega )$ coincides with the set of $q$-analytic functions on $\Omega$ and the representation in Eqn.(1) is unique.

Remark. The operators $(\partial _{\bar {z}})^{q}$ are elliptic for each $q\in \mathbb {Z} _{+}$.Proposition 1 implies ${\mbox{PA}}_{q}(\Omega )\subset C^{\omega }(\Omega )$ for all $q\in \mathbb {Z} _{+}$ (where $C^{\omega }(\Omega )$ denotes the set of real-analytic functions on $\Omega$).

Definition. A function $f$ is called countably analytic on an open subset $U\subset \mathbb {C} ,$ if for every point $p\in U$ there is an open neighborhood $U_{p}$ of $p$ such that on $U_{p}$, $f$ can be represented by a uniformly convergent series $f(z)=\sum _{j=0}^{\infty }a_{p,j}(z){\bar {z}}^{j}$ for holomorphic $a_{p,j}(z)$ on $U_{p}.$

Definition. Let $\Omega \subset \mathbb {C}$ be a domain and let $q\in \mathbb {Z} _{+}$. A polyanalytic function $g(z)=\sum _{j=0}^{q-1}a_{j}(z){\bar {z}}^{j}$, for holomorphic $a_{j}(z)$ on $\Omega$, is called reduced if there exists holomorphic functions $a'_{j}(z)$ such that $a_{j}(z)=z^{j}a'_{j}(z),$ $j=0,\ldots ,q-1.$

Definition in the Case of Complex Analysis of Many Variables

The generalization of $q$-analytic functions to $\mathbb {C} ^{n}$ was introduced properly, in terms of partial differential equation, by Avanissian & Traoré , although it should be mentioned that Balk & Zuev , Parag. 8, gave an alternative (equivalent) definition of the $\alpha$-analytic functions not based upon partial differential equations, and before that Balk introduced such functions for the case of $\mathbb {C} ^{2}$ and proved a generalized a uniqueness theorem for such functions. We present here both definitions and we state their equivalence.

Definition (Balk & Zuev) Let $\omega \subset \mathbb {C} ^{n}$ and $\alpha \in \mathbb {Z} _{+}^{n}.$ A function $f$ on $\Omega$ is called polyanalytic of vectorial order $\alpha$ if it is a polynomial with respect to ${\bar {z}}$ of degree $\alpha _{j}-1$ with respect to ${\bar {z}}_{j}$, $j=1,\ldots ,n$ and with holomorphic coefficients with respect to $z$ on $\omega .$ A function $f$ on a domain $\Omega \subset \mathbb {C} ^{n}$ is called areolar at $p\in \omega$ if it is expressible in some polycylinder $\delta \subset \Omega$ with center $p$ as:

$\sum _{|\beta |=0}^{\infty }({\bar {z}}-{\bar {p}})^{\beta }\phi _{\beta ,p}(z)$

for holomorphic $\phi _{\beta ,p}$ on $\delta .$

Definition ($\alpha$-analytic functions) Let $\Omega \subset \mathbb {C} ^{n}$ be a domain and let $\alpha \in \mathbb {Z} _{+}^{n}.$ A function $f$ on $\Omega$ is called $\alpha$-analytic on $\Omega ,$ if it is a distribution solutions to $\partial _{{\bar {z}}_{1}}^{\alpha _{1}}f=\cdots =\partial _{{\bar {z}}_{n}}^{\alpha _{n}}f=0$ on $\Omega .$ The space of $\alpha$-analytic functions (or polyanalytic functions of order $\alpha$}) on $\Omega$ is denoted by ${\mbox{PA}}_{\alpha }(\Omega )$.

In the last definition, the order of an $\alpha$-analytic function is said to be exact if $f$ has, near each $p_{0}\in \Omega ,$ a representation of the form:

$f(z)=\sum _{0\leq \beta _{j}<\alpha _{j}}a_{\beta }(z){\bar {z}}^{\beta }$

where the $a_{\beta }$ are holomorphic functions near $p_{0},$ such that $a_{\alpha _{j}-1}\not \equiv 0,j=1,\ldots ,n$.

Proposition. Let $\alpha ,\beta \in \mathbb {Z} _{+}^{n}$ ,$\beta _{j}\leq \alpha _{j},$ $j=1,\ldots ,n.$ Let $\Delta (p,r)$ be a polydisc in $\mathbb {C} ^{n},$ with center $p\in \mathbb {C} ,$ where $r=(r_{1},\ldots ,r_{n}).$ Let $f_{1},\ldots ,f_{n}$ be polyanalytic of order $\leq \alpha$ (by which we mean that the separate order with respect to $z_{j}$ is $\leq \alpha _{j},$ $j=1,\ldots ,n$) on $\Delta (p,r)$ such that: $\partial _{z_{j}}^{\beta _{j}}f_{k}=\partial _{z_{k}}^{\beta _{k}}f_{j},\,j,k=1,\ldots ,n$. Then the system $\partial _{z_{j}}^{\beta _{j}}f=f_{j},\quad j=1,\ldots ,n$, has a solution that is polyanalytic of order $\leq \alpha$ on $\Omega$, unique up to addition by a polyanalytic polynomial of order $\leq \alpha$, that is of degree $<\beta _{j}$ with respect to $z_{j}$, $j=1,\ldots ,n.$

Some Basic Properties

Proposition. Let $\Omega \subseteq \mathbb {C}$ be a domain and let $q\in \mathbb {Z} _{+}$. If $f\in {\mbox{PA}}_{q}(\Omega )$ and $f(z)=0$ for all $z\in V$ for some open subset $V\subset \Omega .$ Then $f\equiv 0$ on $\Omega$.

Proposition. Let $q\in \mathbb {Z} _{+},$ let $\Omega \subset \mathbb {C}$ be a domain and let $f\in {\mbox{PA}}_{q}(\Omega ).$ If there exists a point $p_{0}\in \Omega$ such that $\partial _{z}^{j}\partial _{\bar {z}}^{k}f(p_{0})=0$ for all $j,k\in \mathbb {Z} _{\geq 0},$ then $f\equiv 0.$

Proposition. Let $\Omega \subset \mathbb {C}$ be a simply connected domain, let $q\in \mathbb {Z} _{+}$ and let $f\in C^{q}(\Omega )$ be a $q$-analytic function on $\Omega \setminus f^{-1}(0).$ Then $f$ is $q$-analytic on $\Omega .$

Natural Analogues of Polyanalyticity in Various Contexts

Polyanalytic Functions in the Sense of Ahern & Bruna, and $q$-Pseudoanalytic Functions

For further details on the notions presented in this section, see Daghighi, Chapter 15.

Definition 3 ($q$-analytic functions in the sense of Ahern-Bruna in ambient space). Let $\Omega \subset \mathbb {C} ^{n}$ be a domain. A function $f:\Omega \to \mathbb {C}$ is called $q$-analytic in the sense of Ahern-Bruna if $\partial _{{\bar {z}}_{1}}^{\alpha _{1}}\cdots \partial _{{\bar {z}}_{1}}^{\alpha _{n}}f=0$ on $\Omega$ for all multi-integers $\alpha \in \mathbb {N} ^{n},$ such that $\sum _{1\leq j\leq n}\alpha _{j}=q.$

Remark 1. There are many equivalent ways to define differentiable $CR$ functions on a generic embedded $CR$ submanifold $M$ of $\mathbb {C} ^{n}$. Denote by $J$ the complex structure map (i.e. the linear map on $TM$ such that the holomorphic tangent bundle of $M$ is defined fiberwise according to $T_{p}M\cap JT_{p}M$, $p\in M$ (where $J^{2}=-I$). Let $f\in C^{1}(M)$. Then the following are equivalent: (i) $df$ is $\mathbb {C}$-linear on $H^{0,1}M$. (ii) $df$ is $J$-linear on $T^{c}M=TM\cap JTM$. (iii) For all sections $L$ of the holomorphic tangent bundle we have $Lf\equiv 0$ on $M$ (here we can either require that $L$ be any section of the real subbundle $T^{c}M$ or that it is any section of the complex subbundle $H^{0,1}M$). (iv) $f$ can be locally uniformly approximated by entire functions (this is due to the Baouendi-Treves approximation theorem).

Property (iv) has a natural analogue in the $\alpha$-analytic case, and when the $CR$-dimension is $1$, so does property (iii). In higher $CR$ dimension there appears the problem of different choices of basis for the holomorphic tangent bundle so there is no natural decomposition of a section $L$ of $TM\cap JTM$. So from the perspective of property (iii) we may use a generalized version of the notion of $q$-analyticity in the sense of Ahern-Bruna (see Definition 3). One advantage of $q$-analyticity in the sense of Ahern-Bruna (besides the fact that it implies $q$-analyticity along each complex line, see Ahern & Bruna, p.132) is the following.

Proposition. Let $q\in \mathbb {Z} _{+},$ and let $\Omega \subset \mathbb {C}$ be an open subset.Then a function $f\in C^{q}(\Omega )$ is $q$-analytic in the sense of Ahern-Bruna iff $L_{1}\cdots L_{q}f=0$ for any collection $\{L_{1},\ldots ,L_{q}\}$ of sections of $H^{0,1}\Omega ,$ such that each $L_{j}$ is a linear combination of ${\frac {\partial }{\partial {\bar {z}}_{1}}},\ldots ,{\frac {\partial }{\partial {\bar {z}}_{n}}}$ with holomorphic coefficients.

Definition ($q$-analytic functions in the sense of Ahern-Bruna, on $CR$ submanifolds). Let $M\subset \mathbb {C} ^{n}$ be a $C^{q}$-smooth generic $CR$ submanifold of $CR$ dimension $m$. Let $q\in \mathbb {Z} _{+}.$ A function $f:M\to \mathbb {C}$ is called $q$-analytic near $p$ if there is a local basis for $L_{1},\ldots ,L_{m}$ for the set of sections of $H^{0,1}M$ (we will call this a local system of $CR$ vector fields) near $p\in M$, such that we have on an open neighborhood of $p$:

$L_{1}^{\alpha _{1}}\cdots L_{m}^{\alpha _{m}}f=0,\quad \forall \alpha \in \mathbb {N} ^{m}{\mbox{ such that }}\sum _{1\leq j\leq m}\alpha _{j}=q$

It is clear that when $M$ is a complex manifold then the last definition reduces precisely to the definition of $q$-analytic functions in the sense of Ahern-Bruna in ambient space. The definition is locally invariant under local ambient biholomorphic coordinate change. We arrived at this definition from the perspective of (iii) in Remark 1. On the other hand, from the perspective of property (iv) in Remark 1 it is more natural to introduce the following.

Definition ($q$-pseudoanalytic function). Let $M$ be a $C^{q}$-smooth generic $CR$ submanifold in $\mathbb {C} ^{n}$. Let $q\in \mathbb {Z} _{+}.$ A $C^{q}$-smooth function $f:M\to \mathbb {C} ,$ is called $q$-pseudoanalytic at $p_{0}\in M$ if it can be realized, near $p_{0}$ in $M$, as the local uniform limit of ambient $q$-analytic functions (in the sense of Ahern-Bruna), $F_{k}(z),$ $j\in \mathbb {Z} _{+},$ i.e. the $F_{k}$ are defined in an ambient neighborhood of $p_{0}$ in $\mathbb {C} ^{n}.$ $f$ is called $q$-pseudo-analytic on a relatively open subset $U\subseteq M$ if $f$ is $q$-pseudo-analytic at each point of $U.$

Proposition. Let $M$ be a $C^{q}$-smooth generic $CR$ submanifold in $\mathbb {C} ^{n}$. Let $q\in \mathbb {Z} _{+}.$ Then a $C^{q}$-smooth function $f$ is $q$-pseudoanalytic at $p_{0}\in M$ if and only if there exists differentiable $CR$ functions $a_{\beta },$ $\beta \in \mathbb {N} ^{n},$ $\sum _{j}\beta _{j}<q,$ such that $f$ has, near $p_{0}$, the representation:

$f(z)=\sum _{|\beta |<q}a_{\beta }(z){\bar {z}}^{\beta }$ 2

$z\in \mathbb {C} ^{n}\cap M$ near $p_{0}$ (here $z=(z_{1},\ldots ,z_{n})$ and the expressions $f(z),a_{\beta }(z)$ are thus only defined when $z$ lies near $p_{0}$ on $M$ in $\mathbb {C} ^{n}$).

Proposition ($q$-analytic version of the Baouendi & Treves approximation theorem). Let $M\subset \mathbb {C} ^{n}$ be a $C^{q}$-smooth generic $CR$ submanifold of $CR$-dimension $m$, let $q\in \mathbb {Z} _{+}$ and let $f\in C^{q}(M)$. If $f$ is $q$-analytic (in the sense of Ahern-Bruna) on $M$ then $f$ is $q$-pseudoanalytic on $M$.

Proposition. Let $M$ be a $C^{q}$-smooth generic $CR$ submanifold in $\mathbb {C} ^{n}$. Let $q\in \mathbb {Z} _{+}.$ Then $f$ is $q$-analytic on $M$ only if it locally has a representation of the form given by Eqn.(2).

Remark. We stress that the representation in Eqn.(2) is not necessarily unique, as it is well-known that generic $CR$ submanifolds are not sets of uniqueness for $\alpha$-analytic functions in the non-holomorphic case.

Example. In $\mathbb {C} ^{2}$ we consider the flat generic submanifold $M:=\{z\in \mathbb {C} ^{2}:{\mbox{im}}z_{2}=0\}$ that the functions $f(z_{1},z_{2}):=(z_{1}+z_{1}^{2}z_{2})-z_{1}^{2}{\bar {z}}_{2}$, $g(z_{1},z_{2}):=(z_{1}+z_{2})-{\bar {z}}_{2}$. Then $f,g$ are both $q$-analytic in the sense of Ahern-Bruna with $q=2,$ and $f|_{M}=g|_{M}=z_{1}$.

Polyanalytic Functions in Infinite Dimension

For further detail on the result of this section see e.g. the survey of Daghighi, chapter 14. A complex polynomial $P$ in $\mathbb {C} ^{n}$ is called homogeneous of degree $m$ if $P(\lambda z)=\lambda ^{m}P(z)$ for all $\lambda \in \mathbb {C}$, $z\in \mathbb {C} ^{n}$. The local (uniformly convergent) power series of any holomorphic function $f$ can be rewritten as a series of $m$-homogeneous polynomials $f(z)=\sum _{m=0}^{\infty }P_{m}(z),$ where $P_{m}$ is homogeneous of degree $m$, and this is called the homogeneous expansion. If $\phi$ is a linear transformation on $\mathbb {C} ^{n}$ then $P_{m}\circ \phi$ is again homogeneous of order $m$ and the homogeneous expansion of $f$ is given by $\sum _{m}(P_{m}\circ \phi )(z).$ Furthermore, we have for a function $f\in {\mathcal {O}}(\{|z|<1\}),$ (where ${\mathcal {O}}$ denotes the space of holomorphic functions) having a homogeneous expansion $f(z)=\sum _{m=0}^{\infty }P_{m}(z)$ that for $\zeta \in \{|z|=1\}$ and the function of the variable $\lambda \in \mathbb {C} ,$ $|\lambda |<1$ given by $f_{\zeta }(\lambda )=\sum _{m=0}^{\infty }P_{m}(\zeta )\lambda ^{m}$ is holomorphic with induced coefficients $P_{m}(\zeta ).$ If $\Omega \subset \mathbb {C} ^{n}$ is an open subset, $\{P_{m}(z)\}_{m\in \mathbb {N} }$ a sequence where each $P_{m}$ is $m$-homogeneous and if $\sup _{m}|P_{m}(z)|<\infty$ for each $z\in \Omega$ then the series $\sum _{m=0}^{\infty }P_{m}(z)$ converges uniformly on compacts of $\Omega .$ The notion of homogeneous expansion and the idea of the restrictions to each complex line being holomorphic have nice generalization to complex Banach spaces.

There is a well-established field of research on infinite dimensional holomorphy, see e.g. the books of Dineen. and Mujica, Soo Bong Chae and Hervé.

Let $X$ and $Y$ be locally convex vector spaces. Denote by ${\mathcal {L}}(^{m}X,Y)$ the space of $m$-linear mappings from $X^{m}$ (the product space) to $Y,$ and we denote by ${\mathcal {L}}_{s}(^{m}X,Y)$ the vector space of all mappings in ${\mathcal {L}}(^{m}X,Y)$ which are symmetric. o every $\phi \in {\mathcal {L}}(^{m}X,Y)$ (where we do not assume continuity, thus when $Y$ is a scalar field, this is a subset of the algebraic dual) we associate a mapping ${\hat {\phi }}$ defined by ${\hat {\phi }}:=\phi \cdot x^{m},$ and call ${\hat {\phi }}$ the $m$-homogeneous polynomial associated to $\phi .$ Denote by ${\mathcal {P}}(^{m}X,Y)$ the sub-vector space of continuous $m$-homogeneous polynomials. Then the linear mapping from the subspace of continuous functions $\phi \in {\mathcal {L}}(^{m}X,Y)$ to ${\mathcal {P}}(^{m}X,Y),$ defined by $\phi \mapsto {\hat {\phi }},$ is surjective. Furthermore the linear mapping from the subspace of continuous functions in ${\mathcal {L}}^{s}(^{m}X,Y)$ to ${\mathcal {P}}(^{m}X,Y),$ defined by $\phi \mapsto {\hat {\phi }},$ is bijective.

Definition. By a polynomial, we mean a finite sum of elements in $\bigcup _{m}{\mathcal {P}}(^{m}X,Y),$ and we will be considering mainly $Y=\mathbb {C} ,$ and the set of ($\mathbb {C}$-valued continuous) polynomials on $X$ is denoted ${\mathcal {P}}(X).$ We define the norm on ${\mathcal {P}}(^{m}X,Y)$ to be given by $\lVert P\rVert :=\sup _{\lVert x\rVert \leq 1}\lVert P(x)\rVert .$

Definition 4 (See Mujica, p.33). Let $\Omega \subset X$ be open and nonempty, $X$ locally convex. A function $u:X\to Y$ is called holomorphic if$\forall a\in \Omega ,\exists$ a neighborhood $V\subseteq U$ and a sequence of polynomials $\{A_{m}\}_{m\in \mathbb {N} },A_{m}\in {\mathcal {P}}(^{m}U,Y)$ such that: $u(x)=\sum _{m=0}^{\infty }A_{m}(x-a),$ converges uniformly for $x\in V.$The $A_{m}$ are of course members of ${\mathcal {L}}_{s}(^{m}X,Y)$ and if $Y$ is Hausdorff they are uniquely determined by $u.$

Definition. Let $X$ be a separable topological complex Banach space, let $\Omega \subset X$ be an open subset For a map $f:\Omega \to Y$, where $Y$ is a separable Banach space we define for $p\in \Omega$ and $v:=(v_{1},\ldots ,v_{k}),$ $v_{j}\in X,$ $j=1,\ldots ,k$:

${\overline {D}}_{f}(p,v):={\frac {df(p,v)+idf(p,iv)}{2}}$

where $df(p,v_{j})=\lim _{\mathbb {R} \ni \epsilon \to 0}(f(p+\epsilon v_{j})-f(p))/\epsilon .$ For a function $f\in C^{1}$ (i.e. $df(p,v)$ exists for all $p\in \Omega ,$ and each vector $v$ and $(p,v)\mapsto df(p,v)$ is continuous), the function is called Fréchet holomorphic if ${\overline {D}}f\equiv 0$ on $\Omega .$

It is well-known that these functions locally have Frech\'et differentials of all orders at each point and they posses locally convergent homogeneous expansion. Furthermore, the homogeneous polynomials in $x-p_{0}$ in the power series at a point $p_{0}$, which form the terms of the power series, are expressible as multiples of the Fréchet differentials at $p_{0}$ according to the appropriate generalization of the Taylor series, see e.g. Chae, p.392.

Definition. A complex valued functional is Gâteaux holomorphic (or $G$-holomorphic) in a domain $\Omega \subset X$ of a complex Banach space $X$ if it is single-valued and its restriction to an arbitrary analytic plane $\{z:z=p_{0}+\zeta a\}$ ($p_{0}\in \Omega$, $a\in X$, $\lambda$ a complex parameter) is a holomorphic function of $\zeta$ in the intersection of the plane with $\Omega$.

The correspondence $P\leftrightarrow {\stackrel {\vee }{P}}$ establishes an isometric isomorphism. Note that for each $k\in \mathbb {N} ,$ if the differential $d^{k}f(p)$ exists then corresponds to it a natural $k$-homogeneous polynomial which we denote ${\widehat {d^{k}f(p)}}.$ Specifically for $m$-homogeneous polynomials we denote for $j=0,\ldots ,m$:

${\frac {\widehat {d^{j}P}}{j!}}(x):X\ni y\mapsto {\binom {m}{j}}{\stackrel {\vee }{P}}(x^{m-j},y^{j})\in Y$

${\frac {\widehat {d^{j}P}}{j!}}:X\ni x\mapsto {\frac {\widehat {d^{j}P}}{j!}}(x)\in {\mathcal {P}}(^{m}X,Y)$

where we have ${\frac {\widehat {d^{j}P}}{j!}}(x)\in {\mathcal {P}}(^{m}X,Y),$ ${\frac {\widehat {d^{j}P}}{j!}}\in {\mathcal {P}}(^{m-j}X,{\mathcal {P}}(^{j}X,Y))$ and

$P(x+y)=\sum _{j=0}^{m}\left({\frac {\widehat {d^{j}P}}{j!}}(x)\right)(y)$

is just the Taylor expansion of $P$ about $x.$ Understanding polynomials in infinite dimensional complex analysis is more involved than in the finite dimensional case.

Definition. When the $k$:th Fréchet derivative $D^{k}f(a)$ exists the $k$:th differential of $f$ at $a$ is defined as $d^{k}f(a)={\widehat {D^{k}f(a)}}$ i.e. the $k$-homogeneous polynomial corresponding to the $k$:th derivative. For evaluation at $v\in X$ we use the notation $d^{k}f(a)[v]:=D^{k}f(a)[^{k}v].$ For instance we have for $P\in {\mathcal {P}}(^{m}X,Y)$ that $P$ is $C^{\infty }$-smooth and:

$d^{k}P(x)[v]=m(m-1)\cdots (m-k+1){\stackrel {\vee }{P}}(^{m-k}x,^{k}v)$

Thus $D^{k}P:X\to {\mathcal {L}}^{s}(^{k}X,Y)$ is a polynomial map from ${\mathcal {P}}(^{m-k}X,{\mathcal {L}}^{s}(^{k}X,Y)),$ whereas $d^{k}P[v]:X\to {\mathcal {P}}(^{k}X,Y)$ is a polynomial map from ${\mathcal {P}}(^{m-k}X,{\mathcal {P}}(^{k}X,Y)).$

Remark. In an infinite dimensional Banach space a $G$-holomorphic function is not necessarily locally bounded. When the difference quotients along each possible direction converge uniformly then Gâteaux holomorphy at a point implies the existence of the Fréchet derivative at that point. It is known that a function is Fréchet differentiable at each point of an open subset if and only if it is continuous and Gâteaux differentiable at each point, see e.g. Chae, p.392. Also it is known that a function is holomorphic in the sense of Definition 4 if and only if it is continuous and $G$-holomorphic.

Remark. In this text we shall always assume holomorphy without additional specification mean holomorphy in the sense of Definition 4, the space of holomorphic functions on an open set $U$ is denoted ${\mathcal {O}}(U)$. In particular, they will always be $G$-holomorphic and conversely whenever we have a continuous $G$-holomorphic function it will be holomorphic in the sense of Definition 4.

Remark. We mention that there also exists $CR$ functions defined in infinite dimensional complex analysis. Recall that for a $C^{1}$-smooth function $f$ the decomposition into $\mathbb {C}$-linear and $\mathbb {C}$-antilinear parts, $df=\partial f+{\overline {\partial }}f$ implies that $f$ is holomorphic on an open $U\subset \mathbb {C} ^{n}$ iff $df_{p}$ is $\mathbb {C}$-linear on $T_{p}\mathbb {C} ^{n},\forall p\in U.$ Let $X$ be a complex Banach space and $M\subset X$ a subspace both with induced topology and differential structure. $T_{p}X$ itself can be given the structure of a complex Banach space (it can be identified with $X$) namely via the linear map $J_{p}:T_{p}X\to T_{p}X$ i.e. $J_{p}^{2}v=-v,\forall v\in T_{p}X.$ Any vector subspace of $T_{p}X$ which is closed under the application of $J_{p}$ can then be identified as a complex vector space (with induced complex structure from $X$). Let $H_{p}M$ (in some literature this is denoted $T_{p}^{\mathbb {C} }M$ or $T_{p}^{c}M$) denote the largest vector subspace of $T_{p}M$ which is invariant under the application of $J_{p}$ i.e. the largest vector subspace of $T_{p}M$ which under the induced complex structure is a complex vector subspace of $X.$ Kaup (2004) introduced what can be interpreted as solutions to tangential Cauchy-Riemann equations in an infinite dimensional setting, in terms of uniform limits of ambient holomorphic functions.

Definition. Let $X$ be a complex Banach manifold and $M\subset X$ a smooth submanifold. A function $f\colon M\to \mathbb {C}$ is said to satisfy the tangential Cauchy-Riemann equations on $M$ if for all $p\in X,$ the differential $df_{p}\colon T_{p}M\to \mathbb {C}$ is complex linear on the subspace $H_{p}M\subseteq T_{p}M.$ A continuous function $M\to \mathbb {C}$ is to satisfy the tangential Cauchy-Riemann equations on $M$ if it is locally the uniform limit of a sequence of smooth functions that satisfy the tangential Cauchy-Riemann equations on $M$.

Proposition. Let $X$ be a complex Hilbert space (in particular a Banach manifold with a single chart) and $U\subset X$ an open subset. Let $M\subset U$ be a real-analytic hypersurface graphed over its tangent space the sense of Definition \ref{hyperdef}. Then a function $f$ belongs to the space: $\left\{g\in C^{1}(M,\mathbb {C} ):dg_{p}{\mbox{ is }}\mathbb {C} {\mbox{-linear on }}T_{p}^{\mathbb {C} }M,\forall p\in M\right\}$ if and only if $f$ can be locally realized as the uniform limit of ambient holomorphic functions.

We are now ready to introduce a higher order generalization of infinite dimensional holomorphy, this shall be done in terms of a generalization of $q$-analytic functions to an infinite-dimensional setting. For this we use monomials in conjugate variables.

Definition. Let $X$ be a complex Banach space and let ${\mathcal {P}}(^{m}X,\mathbb {C} )$ denote the space of $m$-homogeneous polynomials. Denote:

${\mathcal {P}}_{\mbox{conj}}(^{m}X,\mathbb {C} )=\{\psi :X\to \mathbb {C} {\mbox{ such that }}\psi ={\overline {\phi (z)}},{\mbox{ some }}\phi \in {\mathcal {P}}(^{m}X,\mathbb {C} )\}.$

Definition 5 (Absolute $q$-analytic functions). Let $X$ be a complex Banach space with unconditional (Schauder) basis. A function $f\in C^{q-1}(X,\mathbb {C} )$ is called polyanalytic of order $q$ or $q$-analytic at the origin, if in a neighborhood, $U_{0}$, of the origin in $X,$ $f$ has the representation in terms of a uniformly convergent series:

$f(z)=\sum _{m=0}^{q-1}\sum _{{\mathfrak {m}}_{m}\in B_{m}}a_{{\mathfrak {m}}_{m}}(z){\mathfrak {m}}_{m},a_{{\mathfrak {m}}_{m}}\in {\mathcal {O}}(U_{0})$

where $B_{m}$ is a linearly independent basis for ${\mathcal {P}}_{\mbox{conj}}(^{m}X,\mathbb {C} ).$ $f$ is called countably analytic at $0$ if it has the representation:

$f(z)=\sum _{m=0}^{\infty }\sum _{{\mathfrak {m}}_{m}\in B_{m}}a_{{\mathfrak {m}}_{m}}(z){\mathfrak {m}}_{m}$

and if the required local representation but with translation of the origin holds near every point we simply call $f$, $q$-analytic or countably analytic respectively.

Remark. Let $X$ be a complex Banach space with countable unconditional basis $\{e_{j}\}_{j\in \mathbb {N} }$ (we use here countable basis synonymous with Schauder basis, in particular $X$ is a separable Banach space with a Schauder basis), so any $x\in X$ has a unique representation $x=\sum _{j}c_{j,x}e_{j}$ and this also gives the coordinate functionals $e_{j}':X\to \mathbb {C}$, $e_{j}'(x):=c_{j,x}.$ For each $m\in \mathbb {Z} _{\geq 0},$ let ${\tilde {B}}_{m}$ be a basis for ${\mathcal {P}}(^{m}X,\mathbb {C} )$ (recall that we always assume our homogeneous polynomials to be continuous). By this we mean that for each element $p\in {\mathcal {P}}(^{m}X,\mathbb {C} )$ there exists a sequence $\{c_{j}\}_{j\in \mathbb {N} }$ such that we have a (norm) convergent representation $p(x)=\sum _{j\in \mathbb {N} }c_{j}b_{j},$ where each $b_{j}\in {\tilde {B}}_{m},$ and this representation is unique up to reordering (and since the basis is unconditional reordering does not affect the convergence). We note the following. If $Q\in {\tilde {B}}_{m}$ then, in particular, $Q=Q(x)$ is a function satisfying that $(\mathbf {c} (Q))(x):={\overline {Q(x)}}$ is a function satisfying that $(\mathbf {c} (Q))({\bar {x}})$ is $m$-homogeneous with respect to the variable ${\bar {x}}$ (where ${\bar {x}}$ denotes the complex conjugate $({\bar {x}}_{1},{\bar {x_{2}}},\ldots )$), so we could say that $\mathbf {c} (Q)(x)$ it conjugate-$m$-homogeneous with respect to $x$. This is true independent of the choice of basis $B_{m}.$ Consider a function of the form $p_{m}(x)=\sum _{Q\in B_{m}}a_{Q}(x)(\mathbf {c} (Q))(x)$ where the sum is uniformly convergent and the $a_{Q}$ are holomorphic. It is clear that if $p_{m}$ has such a representation then it will have the analogous representation with any other choice of basis $B_{m}$. Furthermore, we can define $P_{m}(x,y):X\times X\to \mathbb {C}$ according to $P_{m}(x,y):=\sum _{Q\in B_{m}}a_{Q}(x)(\mathbf {c} (Q))({\bar {y}})$. Then $P_{m}$ is separately holomorphic thus by the infinite dimensional version of Hartogs theorem it is a holomorphic function, and satisfies $P_{m}(x,{\bar {x}})=p_{m}(x).$ More specifically, for each fixed $x$, $P(\cdot ,y)$ is $m$-homogeneous with respect to $y$. We shall call a function $P(x,y)$ a pseudopolynomial of degree $m$ if it can be written as $P(x,y)=\sum _{j=0}^{m}P_{j}(x,y)$ for some $P_{j}(x,y):=\sum _{Q\in B_{j}}a_{Q,j}(x)(\mathbf {c} (Q))({\bar {y}})$, where the sum is uniformly convergent and the $a_{Q,j}$ are holomorphic. This can be compared to the finite dimensional case, see e.g. Fritzche & Grauert, p.124 (and Definition \ref{pseudopolynomdef}). Notice that $P$ is uniquely determined by the function $p:=\sum _{j=0}^{m}p_{j}$ and vice versa.

Remark. Evidently, any function having the representation given in Definition 5, will have such a representation independent of the choice of bases $B_{m},$ and furthermore one can equivalently define $f$ to be $q$-analytic if there exists a pseudopolynomial, $F(x,y)$, of degree $m$ with respect to $y$, satisfying $F(x,{\bar {x}})=f(x).$

Proposition. Let $X$ be a complex Banach space with unconditional Schauder basis (which can be viewed as a complex Banach manifold with open unit ball and a single chart) let $U\subset X$ be open and let $f\in C^{q-1}(U,\mathbb {C} ).$ Then $f$ is $q$-analytic on $U$ iff the restriction of $f$ to any one dimensional complex slice is $q$-analytic the sense of Definition 2.

Corollary. $X$ be a complex Hilbert manifold and let $f$ and $g$ be two $q$-analytic functions on a domain $U\subset X$. If $f=g$ on an open subset $E\subset U,$ then $f\equiv g$ on $U.$

Proposition. Let $X$ be a complex Banach space with countable basis and let $f$ be a $C^{q}(U,\mathbb {C} )$-smooth function on an open neighborhood $U$ of $0$ in $X$which is $q$-analytic on $U\setminus f^{-1}(0)$. Then $f$ is $q$-analytic on $U.$

$q$-analyticity in Hypercomplex Analysis

The defining equations for polyanalytic functions have natural counterparts in the more general theory of Clifford algebras, but we shall here begin with the specific case of $\mathbb {R} ^{4}$ so that we can give an account of the original pioneering work of Brackx,

Definition. Denote by $\mathbb {H}$ the algebra of real quaternions with standard multiplication $\times$ and componentwise addition "$+$", recall that $q\in \mathbb {H}$, if $q=q_{0}+q_{1}i+q_{2}j+q_{3}k,$ where $1,i,j,k$ are basis elements and the multiplication is the one induced by $i^{2}=j^{2}=k^{2}=ijk=-1$ (and satisfies $ij=k=-ji,$ $jk=i=-kj,$ $ki=j=-ik$). The identity is given by the real quaternion $1$ and the real quaternions form the center of the algebra, i.e. they are precisely the elements that commute with all members of the algebra. For $q\neq 0,$ we have an inverse $q^{-1}$ given by $q^{-1}=(q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2})(q_{0}+q_{1}i+q_{2}j+q_{3}k),$ i.e. $\mathbb {H}$ is a non-commutative but associative division algebra over $\mathbb {R} .$ Note that $q=(q_{0}+q_{1}i)+(q_{2}+q_{3}i)j=:z+wj,$ with 'conjugate' given by ${\bar {q}}:=q_{0}-q_{1}i-q_{2}j-q_{3}k=(q_{0}-q_{1}i)-j(q_{2}-q_{3}i)={\bar {z}}-j{\bar {w}}$ and $q{\bar {q}}={\bar {q}}q=|q|^{2}=q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2}=|z|^{2}+|w|^{2}.$

Definition. Let $M$ be a $4$-dimensional, differentiable, oriented manifold, $M\subset \Omega$ for an open non-empty subset $\Omega \subset \mathbb {R} ^{4}.$ Let $p\in \{1,2,3,4\},$ and let $C$ be a $p$-chain on $M.$ Let $\omega _{\alpha }$, $\alpha =0,1,2,3,$ be real $p$-forms on $M$ i.e. $\omega _{\alpha }=\sum _{h}\eta _{\alpha ,h}dx^{h},$ where $h=(h_{1},\ldots ,h_{p})\in \{0,1,2,3\}^{p},$ $0\leq h_{1}<\cdots <h_{p}\leq 3,$, $dx^{h}\in \Lambda ^{p}w,$ where $w$ is the $4$-dimensional vector space with basis $\{dx_{0},dx_{1},dx_{2},dx_{3}\}$ and $\eta _{\alpha ,h}\in C^{r}(\Omega ),$ $\eta _{\alpha ,h}:\mathbb {R} ^{4}\to \mathbb {R} ,$ for all $\alpha ,h.$ Let $\{e_{0},e_{1},e_{2},e_{3}\}$ denote the basis of the algebra $\mathbb {H}$. Then we can decompose each quaternion-$p$-form $\omega$ as $\omega =\sum _{\alpha =0}^{3}e_{\alpha }\omega _{\alpha }.$ Define:

$\int _{C}\omega =\sum _{\alpha =0}^{3}e_{\alpha }\int _{C}\omega _{\alpha }$

A function $f:\mathbb {R} ^{4}\to \mathbb {H}$ can be represented by:

$f=\sum _{\alpha =0}^{3}e_{\alpha }f_{\alpha },\quad x=(x_{1},x_{2},x_{3},x_{4})\mapsto \sum _{\alpha =0}^{3}e_{\alpha }f_{\alpha }(x)$

where $f_{\alpha }$, $\alpha =0,1,2,3,$ are real-valued. Define:

$D:=\sum _{\beta =0}^{3}e_{\beta }{\frac {\partial }{\partial x_{\beta }}}$

so that:

$Df=\sum _{\alpha ,\beta =0}^{3}e_{\beta }e_{\alpha }{\frac {\partial f_{\alpha }}{\partial x_{\beta }}},\quad (fD)=\sum _{\alpha ,\beta =0}^{3}e_{\alpha }e_{\beta }{\frac {\partial f_{\alpha }}{\partial x_{\beta }}}$

A function $f:\mathbb {R} ^{4}\to \mathbb {H}$ is called left (right) $k$-monogenic on $\Omega$ if (i) $f_{\alpha }\in C^{k}(\Omega ),\alpha =0,1,2,3$ (this shall be denoted $f\in C^{k}(\Omega )$) and (ii):

$D^{k}f=D(D^{k-1}f)=0{\mbox{ in }}\Omega \quad ((fD^{k}=0){\mbox{ in }}\Omega )$

Define further:

${\overline {D}}:=\sum _{\beta =0}^{3}\epsilon _{\beta }e_{\beta }{\frac {\partial }{\partial x_{\beta }}},\quad \epsilon _{0}=1,\epsilon _{\beta }=-1,\beta =1,2,3$

Not only are the defining equations analogous to those of $q$-analytic functions, but the relation:

$D{\overline {D}}={\overline {D}}D=\Delta e_{0},\qquad \Delta :=\sum _{\beta =0}^{3}e_{\beta }{\frac {\partial ^{2}}{\partial x_{\beta }^{2}}}$

immediately implies:

$D^{k}{\overline {D}}^{k}f={\overline {D}}^{k}(D^{K}f)=\Delta ^{k}f$

thus we obtain the following counterpart to the fact that $q$-analytic functions have $q$-harmonic real and imaginary parts, thus are complex-valued $q$-harmonic functions.

Proposition. If $f\in C^{2k}(\Omega )$ is left (right) $k$-monogenic on $\Omega$ then $f$ is $k$-harmonic in the sense that $\Delta ^{k}f=({\overline {D}}D)^{k}f=\sum _{\alpha =0}^{3}{\frac {\partial ^{2}}{\partial x_{\alpha }^{2}}}f=0$ on $\Omega .$

We denote $d{\hat {x}}_{0}:=dx_{1}\wedge dx_{2}\wedge dx_{3}$, $d{\hat {x}}_{1}:=dx_{0}\wedge dx_{2}\wedge dx_{3},$ $d{\hat {x}}_{2}:=dx_{0}\wedge dx_{1}\wedge dx_{3}$ ,$d{\hat {x}}_{3}:=dx_{1}\wedge dx_{1}\wedge dx_{2}$, $d\sigma _{x}:=\sum _{\alpha =0}^{3}(-1)^{\alpha }e_{\alpha }d{\hat {x}}_{\alpha }.$

Theorem (Quaternion version of the Cauchy integral formula) Let $\Omega \subset \mathbb {R} ^{4}$ be an open subset, let $f$ be a left-k-monogenic function on $\Omega$ and let $S\subset \Omega$ be a $4$-dimensional, compact, differentiable, oriented manifold with boundary. Then for each $x\in {\stackrel {\circ }{S}}$:

$f(x)=\int _{\partial S}\sum _{j=0}^{k-1}(-1)^{j}g_{j+1}(u-x)d\sigma _{u}D^{j}f(u)=$${\frac {1}{2\pi ^{2}}}\int _{\partial S}\sum _{j=0}^{k-1}(-1)^{j}{\frac {{\bar {u}}-{\bar {x}}}{|u-x|^{4}}}{\frac {(u_{0}-x_{0})^{j}}{j!}}d\sigma _{u}D^{j}f(u)$where $|u-x|$ denotes the Euclidean distance in $\mathbb {R} ^{4},$ ${\mbox{dist}}(x,u).$

An Analogue of polyanalyticity in the Case of Clifford Algebras

Definition (Clifford algebra). Let $K=\mathbb {R}$, let $V$ be a finite-dimensional $K$-vector space and let $\xi :V\times V\to K$ be a symmetric bilinear form with associated quadratic form $Q:V\to K$, $Q(x)=\xi (x,x).$ A Clifford algebra, ${\mbox{Cliff}}(V,Q)$, is a unital associative algebra together with a linear map $\nu :V\to {\mbox{Cliff}}(V,Q)$ satisfying:

(1) $(\nu (x))^{2}=Q(x)\cdot 1,$ (where $1$ denotes the multiplicative identity in the algebra) for all $x\in {\mbox{Cliff}}(V,Q).$

(2) Given any unital associative algebra $A$ over $K$ and any linear map $\eta :V\to A$ such that $(\eta (x))^{2}=Q(x)\cdot 1_{A}$ for all $x\in V$ (where $1_{A}$ denotes the multiplicative identity in $A$), there is a unique algebra homomorphism $\phi :{\mbox{Cliff}}(V,Q)\to A$(i.e. $\phi$ is a linear map that is also a ring homomorphism such that $\phi (1_{\mbox{Cliff}}=1_{A}$) such that $\phi \circ \nu =\eta$.

Remark. A quadric form, $Q$, on an $n$-dimensional $\mathbb {R}$-vector space, $V,$ can be represented as $Q(x)=\sum _{j=1}^{n}\sum _{k=1}^{n}q_{jk}x_{j}x_{k},$ which can be identified in matrix notation as $x^{T}[Q]x,$ where $x^{T}$ denotes the transpose vector and $[Q]:=[q_{jk}]_{jk}$ a symmetric matrix. The signature (the triplet of the number of positive,zero,negative eigenvalues respectively), $g,$ of $[Q]$ is a parameter upon which the Clifford algebra ${\mbox{Cliff}}(V,Q)$ depends. A quadric form is called non-degenerate if $[Q]$ has no zero eigenvalues. Note that necessarily the sum of the elements of the signature is $n.$ In this text we shall consider only the case of so-called standard Clifford algebras over a finite dimensional $\mathbb {R}$-vector space, $V,$ with orthonormal basis $\{e_{1},\ldots ,e_{n}\},$ by which we mean that ${\mbox{Cliff}}(V,Q)$ is generated by the basis elements and the non-degenerate quadric form $Q$ of signature $(0,0,n)$ chosen to induce satisfies the conditions $e_{k}^{2}=-1,$ $e_{k}e_{j}=-e_{j}e_{k},$ $k\neq j.$ Any element $x$ in the standard Clifford algebra can be written $x=\sum _{J}c_{J}e_{J}$ where $J=\{i_{1},\ldots ,i_{k}\}$ is any subset (possibly empty) of $\{1,\ldots ,n\}$ and $i_{1}<\cdots <i_{k},$ and $e_{J}:=e_{i_{1}}\cdots e_{i_{k}},$ $e_{\emptyset }:=1,$ the identity element. So the dimension over $\mathbb {R}$ is $\sum _{k=0}^{n}{\binom {n}{k}}=2^{n}.$ In the case of non-degenerate $Q$, sometimes authors denote by $\mathbb {R} _{0,n}$ the standard Clifford algebra associated to $\mathbb {R} ^{n}$ and $Q$ a nondegenerate quadric form with signature $(0,0,n)$.

Definition. A complex algebra (not necessarily commutative) $A$ is called a $H^{*}$-algebra if it is equipped with an inner product $(,)$ and an involution $x\to {\overline {x}}$ such that (i) $(xy,z)=(y,{\overline {x}}z).$ (ii) $(yx,z)=(y,z{\overline {x}})$ for all $x,y,z\in A$ and (iii) $A$ is a Banach algebra for the norm $\lVert \cdot \rVert _{0}$ induced by the inner product.

Definition. Let $A$ be a finite dimensional (standard universal) Clifford algebra with basis $\{e_{1},\ldots ,e_{n}\}.$ The subspace spanned by the ${\binom {n}{p}}$ products $e_{\alpha },$ cardinality of $\alpha$ equal to $p,$ the subspace will be denoted $A_{p}.$ Let $\lambda =\sum _{\alpha }e_{\alpha }\lambda _{\alpha }$ be a Clifford number. The coefficient $\lambda _{\alpha }$ of $e_{\alpha }$ will be denoted $[\lambda ]_{0}$ The number $[\lambda ]_{0}$ is called the scalar part of $\lambda .$ An inner product in $A$ is defined by putting for all $a,b\in A$:

$(a,b)_{0}=2^{n}[a,{\bar {b}}]_{0}=2^{n}\sum _{\alpha }\lambda _{\alpha }b_{\alpha }$

Remark. Note that:

$(a,b)_{0}=(b,a)_{0}=({\bar {a}},{\bar {b}})_{0}=({\bar {b}},{\bar {a}})_{0}$

$|\lambda |_{0}={\sqrt {(\lambda ,\lambda )_{0}}}=2^{\frac {n}{2}}{\sqrt {[\lambda ,{\bar {\lambda }}]_{0}}}2^{\frac {n}{2}}{\sqrt {\sum _{\alpha }\lambda _{\alpha }^{2}}}$

In this way $A$ is a real Hilbert space and a Banach algebra with $|ab|\leq |a|_{0}|b|_{0}.$

Proposition.The finite dimensional Clifford algebra $A$ equipped with inner product $(\cdot ,\cdot )$ and involution $a\mapsto {\overline {a}}$ is a finite dimensional $H^{*}$-algebra.

Let $m\leq n,$ $m\neq 0$ and let $\Omega \subset \mathbb {R} ^{m+1}$ be a nonempty open subset. Denote by $M_{k}(\Omega ,A)$ the set of functions $f\in C^{k}(\Omega ,A)$ such that $D^{k}f=0$ on $\Omega ,$ where $k\in \mathbb {N}$ and $D:=\sum _{i=0}^{m}e_{i}\partial _{x_{i}}.$ $D$ is the hypercomplex generalization of the Cauchy-Riemann operator and when $n=m=1$ the solutions to $D^{k}f=0$ on $\Omega$ are precisely the $k$-analytic functions. For general $n,m,k$ the space $M_{k}(\Omega ,A)$ is the subspace of the set of $\mathbb {R} ^{2^{n}}$-valued $k$-harmonic functions. It is known that $M_{k}(\Omega ,A)$ equipped with the topology of uniform convergence on compacts, is a right $A$-Fréchet module. Since dim$A=2^{n}$ the equation $D^{k}f=0$ is equivalent to a system of $2^{n}$ linear partial differential equations, each of order $k$, in the unknown real valued functions $f_{\alpha }.$ If the basis elements $e_{\alpha }$ of $A$ are ordered in a certain way, then the left regular representation of $A$ allows us to associate to each $\lambda \in A$ a $2^{n}\times 2^{n}$ real matrix $\Theta (\lambda ).$ Since $A$ has an identity this representation is an isomorphism. Setting ${\overline {D}}=\sum _{i=0}^{m}{\overline {e}}_{i}\partial _{x_{i}}=e_{0}\partial _{x_{0}}-\sum _{j=1}^{m}e_{j}\partial _{x_{j}}$ and $\Delta =\sum _{i=0}^{m}\partial _{x_{i}}^{2}$ we have $D{\overline {D}}-{\overline {D}}D=\Delta e_{0}$.

Proposition. The system of differential equations associated to the hypercomplex differential operator $D^{k}$ is strongly elliptic.

Polyanalytic Functions on subsets of $\mathbb {Z} [i]$

Some of the pioneers of the investigation of analogues of complex analytic functions on $\mathbb {Z} [i]$ were Isaacs.,, Ferrand and Duffin . Notable modern contributions to the field include the works of Kiselman ,. In Isaacs the monodiffric functions of the first kind on the discrete complex plane, where defined square-wise as those that where annihilated by a certain first order linear difference operator, in particular a complex-valued function $f$ on $\mathbb {Z} [i]$, is monodiffric of the first kind on a square with vertices $\{z,z+1,z+i,z+i+1\},$ whose lower left point is $z\in \mathbb {Z} [i],$ if and only if $f$satisfies:

$f(z+1)-f(z)={\frac {f(z+i)-f(z)}{i}}$ 3

We shall say that $f$ is monodiffric of the first kind at $z$ if and only if $f$ satisfies equation \ref{firstkind}.We shall say that a function $f$ in the discrete complex plane is monodiffric of the second kind at $z\in \mathbb {Z} [i],$ if and only if:

${\frac {f(z+1+i)-f(z)}{i+1}}={\frac {f(z+i)-f(z+1)}{i-1}}$ 4

Ferrand (who uses a discrete version of Moreras theorem) used the term preholomorphic for the monodiffric functions of the second kind. In this paper we shall say that a function $f$ in the discrete complex plane is monodiffric functions of the third kind at $z\in \mathbb {Z} [i]$ if and only if:

$f(z+1)-f(z-1)={\frac {f(z+i)-f(z-i)}{i}}$ 5

The monodiffric functions of the third kind (these where also introduced by Isaacs, p.179) appear less frequently in the literature, and then they are not referred to as monodiffric functions of the third kind. We shall be interested in powers of the operators in Eqn.(3), Eqn.(4), and Eqn.(5) respectively. To avoid confusion we point out that in Kurowski, the functions that we here call monodiffric of the third kind, are called monodiffric of the second kind.

Remark. Kiselman defines a polygon determined by the ordered set $(a_{0},\ldots ,a_{N}),$ $a_{j}\in \mathbb {Z} [i],$ $j=0,\ldots ,N$ to be a 4-curve if $a_{j}-a_{j-1}\in \{\pm 1,\pm i\}$, $j=1,\ldots ,N$ and it is a well-known result see e.g. Isaacs, p.183, that if $f$ is a monodiffric function of the first kind then:

$\int _{\gamma }f(z)dz=0$

for each closed (non-self-intersecting) 4-curve $\gamma$. The corresponding result for monodiffric functions of the second kind also holds true (see e.g.\ Duffin, Corollary 2.1.1).

We have chosen not to adapt that terminology and instead use the terminology used by e.g. Kiselman,, regarding monodiffric functions of the first and second kind, see also Daghighi.

Definition ($q$-polyanalytic functions; polyanalytic functions of order $q$) Define for complex-valued functions $f$ on $\mathbb {Z} [i]$:

$L_{1}f(z):=f(z+1)-f(z)+i(f(z+i)-f(z))$

$L_{2}f(z):=f(z)+if(z+1)-f(z+1+i)-if(z+i)$

$L_{3}f(z):=f(z+1)-f(z-1)+i(f(z+i)-f(z-i))$

We define, for a given positive integer $q$, and a fixed $j\in \{1,2,3\},$ a complex-valued function $f\colon \mathbb {Z} [i]\to \mathbb {C}$ to be:

$q$-polyanalytic (or polyanalytic of order $q$) of the first kind at $z\in \mathbb {Z} [i]$ if:

$L_{1}^{q}f(z)=0$

$q$-polyanalytic (or polyanalytic of order $q$) of the second kind at $z\in \mathbb {Z} [i]$ if:

$L_{2}^{q}f(z)=0$

and $q$-polyanalytic (or polyanalytic of order $q$) of the third kind at $z\in \mathbb {Z} [i]$ if:

$L_{3}^{q}f(z)=0$

If the condition holds true at each point of a subset $S\subseteq \mathbb {Z} [i]$ where the defining operator is defined, then we say that $f$ is $q$-polyanalytic (or polyanalytic of order $q$) of the first, second or third kind, respectively on $S$ and when it is clear from the context what $S$ is we simply say that $f$ is $q$-polyanalytic (or polyanalytic of order $q$) of the first, second or third kind respectively.

Kiselman, Sec 3, and Kurowski , p.1, pointed out that at the level of ideas, the operators defined by Eqn.(3) and Eqn.(4) are quite similar. It is clear however that the solution spaces defined by the operators $L_{2},L_{3}$ are not equivalent. Various characterizations of polyanalytic of order $q$ on subsets of $\mathbb {Z} [i]$ can be found in Daghighi

A motivation for the definition of $q$-polyanalytic functions of the third kind

Definition (Gaussian structure). Let $G$ be an additive abelian group. Also equip $G\times G$ with the additive group structure:

$(p_{1},p_{2})+(q_{1},q_{2}):=(p_{1}+q_{1},p_{2}+q_{2})$

$(p_{1},p_{2}),(q_{1},q_{2})\in G\times G.$ Define for each $(v_{1},v_{2})\in G\times G$:

${\mathcal {J}}:=(v_{1},v_{2})\mapsto (-v_{2},v_{1}).$

Let $G$ also be a directed graph with adjacancy relation $\sim _{G}$. Define an extension, $\sim ,$ of the adjacancy relation $\sim _{G},$ by defining for any pair of points $p,q\in G\times G$ such that, $p=(p_{1},p_{2}),$ $p\neq q$: $q\sim p$ $\Longleftrightarrow$ $p=q+{\mathcal {J}}^{j}((s_{1}-p_{1},0))$ for some $j\in \mathbb {Z} _{\geq 0},$ and some $s_{1}\sim _{G}p_{1}.$ The structure, ${\mathcal {G}},$ so obtained is called the Gaussian structure induced by $G$. When $G=\mathbb {Z}$, we shall denote the Gaussian structure by ${\mathcal {G}}_{\mathbb {Z} }.$

It is clear that letting $G=\mathbb {Z}$ with adjacancy being determined by ($q\sim p$, $q\neq p$) $\Longleftrightarrow$ ($p\in \{q\pm 1\}$), and $\mathbb {Z} ^{2}$ assumed to have the natural addition induced by $\mathbb {Z}$, we obtain a Gaussian structure which aside from its graph properties, can, when equipped with the usual multiplication, be identified with $\mathbb {Z} [i].$ Indeed, we have $G\times G=\mathbb {Z} ^{2}$, and the map ${\mathcal {J}}$ can be identified with 90 degree clockwise rotation in the plane. However, we are introducing graph properties (which are not a priori part of the definition of the Gaussian structure induced by $\mathbb {Z}$) which in the particular example of $\mathbb {Z} [i],$ implies $z\sim w,$ and $z\neq w,$ then $z_{1}=w_{1}\pm i$ or $z_{2}=w_{2}\pm 1$ and each point has precisely four adjacent points except itself. We may obviously introduce multiplication $(z_{1},z_{2})\cdot (w_{1},w_{2}):=(z_{1}w_{1}-z_{2}w_{2},z_{1}w_{2}+z_{2}w_{1}),$ and thus be able to identify the Gaussian structure, ${\mathcal {G}}_{\mathbb {Z} },$ induced by $G=\mathbb {Z}$ with $\mathbb {Z} [i]$ but with additional graph structure as above.

Definition ($q$-polyanalytic functions of the third kind on Gaussian structures). Let $q\in \mathbb {Z} _{+}$ and let ${\mathcal {G}}$ be a Gaussian structure induced by a group $G$ (in particular we have an adjacancy relation $\sim$ on $G\times G$). Since $G$ is directed we can assign to each ordered pair of adjacent points, $s,t,$ $\lambda _{s,t}=1$ ($\lambda _{s,t}=-1$) if the ordered pair is of positive (negative) direction. We define a complex-valued function $f\colon {\mathcal {G}}\to \mathbb {C}$ to be $q$-polyanalytic of the third kind at $p\in {\mathcal {G}}$ if and only if, $L_{3}f(p)=0,$ where $L_{3}f(p)=i\sum _{q\sim p,q_{2}\neq p_{2}}f(q)\cdot \lambda _{p_{2},q_{2}}+\sum _{q\sim p,q_{1}\neq p_{1}}f(q)\cdot \lambda _{p_{1},q_{1}}.$

In the case where the inducing group $G$ is $\mathbb {Z}$ the other two kinds of $1$-polyanalytic functions have equally natural formulations.

Definition. Let $q\in \mathbb {Z} _{+}$ and let ${\mathcal {G}}_{\mathbb {Z} }$ be the Gaussian structure induced by $\mathbb {Z}$. We define a complex-valued function $f\colon {\mathcal {G}}\to \mathbb {C}$ to be {\ $q$-polyanalytic of the $j$:th kind} at $z\in {\mathcal {G}}$ if and only if, $L_{j}^{q}f(z)=0$, $j=1,2,3,$ where $L_{1}f(z):=f(z+1)-f(z)+i(f(z+i)-f(z))$, $L_{2}f(z):=f(z+1)-f(z-1)+i(f(z+i)-f(z-i))$, $L_{3}f(z):=f(z+1+i)-f(z)+if(z+i)-if(z+1).$ If the condition holds true at each point of a subset $S\subseteq {\mathcal {G}}_{\mathbb {Z} }$ where the defining operator is defined, then we say that $f$ is $q$-polyanalytic of the $j$:th kind on $S$ and when it is clear from the context what $S$ is we simply say that $f$ is $q$-polyanalytic of the $j$:th kind.

Recall that if $M$ is an $n$-dimensional smooth real manifold and $p\in M,$ then we can define the set of tangent vectors at $p$ (or tangent space at $p$) as the set of vectors $v$ such that there exists a differentiable curve $\gamma \colon (-\epsilon ,\epsilon )\to M,$ some $\epsilon >0,$ $\gamma (0)=p,$ such that $v={\frac {\partial \gamma }{\partial t}}(0),$ and acts on the set of differentiable functions, defined on a neighborhood of $p$, according to$v(g):={\frac {\partial (f\circ \gamma )}{\partial t}}(0),$ for differentiable $f\colon U\to \mathbb {C} ,$ $p\in U$, $U$ an open neighborhood of $p$ in $M.$ The tangent space at $p$ is denoted $T_{p}M.$ Also for differentiable $f\colon M\to \mathbb {C} ,$ we define the differential map $d_{p}f\colon T_{p}M\to \mathbb {C} ,$ as $d_{\gamma (0)}f({\frac {\partial \gamma }{\partial t}}(0))={\frac {\partial (f\circ \gamma )}{\partial t}}(0).$

Definition. Let ${\mathcal {G}}$ be a graph and let $p\in {\mathcal {G}}$. A path $\Gamma$ through $p$ in $G$ is an ordered set of points $\Gamma (j)=z_{j}\in G,$ $j=-m_{1},\ldots ,m_{2},$ for nonnegative integers $m_{1},m_{2},$ such that $z_{j}\sim z_{j+1},$ $j=-m_{1},\ldots m_{2}-1$, and $p\in \{\Gamma (j),j=-m_{1}+1,\ldots ,m_{2}-1\}.$ When the base point is not essential to the argument being made we shall simply use the term path in ${\mathcal {G}}$. For each $p\in {\mathcal {G}},$ denote $T_{p}{\mathcal {G}}=\{v\in G\colon v=q-p,\,q\sim p\}.$ This is the set of tangents. Obviously, the cardinality of $T_{p}{\mathcal {G}}$ may vary dependent upon the base point $p.$ Let $f$ be a map ${\mathcal {G}}\to {\mathcal {D}},$ for an additive abelian group ${\mathcal {D}}.$For each $p\in {\mathcal {G}},$ we have a map $d_{p}f\colon T_{p}G\to {\mathcal {D}},$ according to $v=(q-p)\mapsto f(q)-f(p).$ So there exists a path $\Gamma$ containing $p$ and $q$ such that $d_{p}f(v)=f(\Gamma (j_{0}+1))-f(p)$ where $\Gamma (j_{0})=0.$

Definition ($1$-polyanalytic functions of the third on Gaussian structures). Let ${\mathcal {G}}$ be the Gaussian structure induced by $G,$ where $G$ is an additive group. Let $R$ be an additive abelian group and let $f$ be a function ${\mathcal {G}}\to R^{2},$ where $R^{2}$ is equipped with the componentwise addition. $f$ is called a $1$-polyanalytic function of the third kind (with respect to the Gaussian structure ${\mathcal {G}},$ at $p$), if (using the notation of Definition \ref{ptdef}) we have:

$d_{p}f(v)-d_{p}f(-v)+{\mathcal {J}}'(d_{p}f({\mathcal {J}}v)-d_{p}f(-{\mathcal {J}}v))=0,v\in T_{p}{\mathcal {G}}$

Where ${\mathcal {J}}',$ is defined by ${\mathcal {J}}'(A,B)=(-B,A),$ and ${\mathcal {J}}(v_{1},v_{2})=(-v_{2},v_{1})$.

From the definitions it is clear that this coincides with the case of $1$-polyanalytic functions of the third kind from Definition \ref{qanaldef}, when e.g. $R=\mathbb {R} ,$ $G=\mathbb {Z}$.

External links

• A. Daghighi, Theory of Polyanalytic Functions

Category:Complex Analysis Elliptic partial differential equations