arXiv subjects

Abtin Daghighi

Publications and source records attributed to Abtin Daghighi.

5 recordsLinked to original sources

arXiv2025

Theory of Polyanalytic functions

Brief Description: The book provides a unique highly self-contained text introducing the reader to the classical and modern theory of polyanalytic functions and their generalizations. This is a subbranch of complex analysis of several variables, in particular polyanalytic functions where introduced as a natural generalization of holomorphic functions. The book includes a solid treatment of the most important well-known results from the inception of the subspecialty, but also covers a wide variety of generalizations and recent developments which have arisen since the latest monograph on the subject was published. A careful selection has been made of topics and results to include, in order to provide the reader with a classic repertoire and a modern overview. The style and presentation is old-school and rigorous and the author does not shy away from presenting complete proofs also for important theorems which require involved background for a modern demonstration.

math.CV↗

arXiv2021

On Radó's theorem for polyanalytic functions

We prove versions of Radó's theorem for polyanalytic functions in one variable and also on simply connected $\mathbb{C}$-convex domains in $\mathbb{C}^n$. Let $Ω\subset \mathbb{C}$ be a bounded, simply connected domain and let $q\in \mathbb{Z}_+.$ Suppose at least one of the following conditions holds true: (i) $g\in C^{q}(Ω).$ (ii) $g\in C^κ(Ω),$ for $κ=\min\{1,q-1\},$ such that $g$ is $q$-analytic on $Ω\setminus g^{-1}(0)$ and such that Re$g$ (Im$g$ respectively) is a solutions to the $p'$-Laplace equation ($p''$-Laplace equation respectively) on $Ω\setminus g^{-1}(0)$, for some $p',p''>1$. Then $g$ agrees (Lebesgue) a.e.\ with a function that is $q$-analytic on $Ω.$ In the process we give a simple proof of the fact that: If $f\in C^q(Ω)$ is $q$-analytic on $Ω\setminus f^{-1}(0)$ then $f$ is $q$-analytic on $Ω.$ The extensions of the results to several complex variables are straightforward using known techniques.

math.CV↗

arXiv2019

An algebra of polyanalytic functions

The most important uniform algebra is the family of continuous functions on a compact subset $K$ of the complex plane $\mathbb{C}$ which are analytic on the interior int$(K)$ For compact sets $K$ which are regular (i.e. $K =$int$(K)$ and for polyanalytic functions, we introduce analogous spaces, which are Banach spaces with respect to the sup-norm, but are not closed with respect to the usual pointwise multiplication. We shall introduce a multiplication on these spaces and investigate the resulting algebras.

math.CV↗

arXiv2018

Level sets of certain classes of $α$-analytic functions

For an open set $V\subset\mathbb{C}^n$, denote by $\mathscr{M}_α(V)$ the family of $α$-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded domain $Ω\subset \mathbb{C}^n$, with continuous boundary (that in each variable separately allows a solution to the Dirichlet problem), a function $f \in \mathscr{M}_α(Ω\setminus f^{-1}(0))$ automatically satisfies $f\in \mathscr{M}_α(Ω)$, if it is $C^{α_j-1}$-smooth, in the $z_j$ variable, $α\in \mathbb{Z}^n_+$, up to the boundary. For a submanifold $U\subset \mathbb{C}^n$, denote by $\mathfrak{M}_α(U)$ the set of functions locally approximable by $α$-analytic functions where each approximating member and its reciprocal (off the singularities) obey the boundary maximum modulus principle. We prove, that for a $C^3$-smooth hypersurface, $Ω$, a member of $\mathfrak{M}_α(Ω)$, cannot have constant modulus near a point where the Levi form has a positive eigenvalue, unless it is there the trace of a polyanalytic function of a simple form.

math.CV↗

arXiv2015

A note on a conjecture concerning boundary uniqueness

We consider the following conjecture (from Huang, et al): Let $Δ^+$ denote the upper half disc in $\mathbb{C}$ and let $γ= ( - 1, 1)$ (viewed as an interval in the real axis in $\mathbb{C}$). Assume that $F$ is a holomorphic function on $Δ^+$ with continuous extension up to $γ$ such that $F$ maps $γ$ into $\{|\mbox{Im} z|\leq C|\mbox{Re} z|\},$ for some positive $C.$ If $F$ vanishes to infinite order at $0$ then $F$ vanishes identically. We show that given the conditions of the conjecture, either $F\equiv 0$ or there is a sequence in $Δ^+$, converging to $0,$ along which $\mbox{Im} F/\mbox{Re} F$ (defined where $\mbox{Re} F\neq 0$) is unbounded.

math.CV↗