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Theory of Polyanalytic functions

Abtin Daghighi

TL;DR

The work develops a cohesive, multi- viewpoint theory of polyanalytic functions, unifying q-analytic and α-analytic frameworks through PDE, functional-analytic, and geometric perspectives. It establishes local representations $f(z)=\sum_{j=0}^{q-1} a_j(z)\bar{z}^j$, proves ellipticity/hypoellipticity, and connects single-variable and several-variable theories via AB and areolar viewpoints. A central thread is the zero-set and boundary-value theory, leveraging Schwarz functions, Pompieu-type integrals, and Nevanlinna tools to obtain identity principles, extension theorems, and a polyanalytic analogue of the fundamental theorem of algebra, plus Radó-type results. Together these form a robust foundation for further hypoanalytic/CR-geometry developments and potential applications in complex analysis and several complex variables. The exposition emphasizes pure mathematics, rigorous proofs, and a spectrum of classical-to-modern techniques, charting trajectories for ongoing advances in polyanalytic function theory.

Abstract

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.

Theory of Polyanalytic functions

TL;DR

The work develops a cohesive, multi- viewpoint theory of polyanalytic functions, unifying q-analytic and α-analytic frameworks through PDE, functional-analytic, and geometric perspectives. It establishes local representations , proves ellipticity/hypoellipticity, and connects single-variable and several-variable theories via AB and areolar viewpoints. A central thread is the zero-set and boundary-value theory, leveraging Schwarz functions, Pompieu-type integrals, and Nevanlinna tools to obtain identity principles, extension theorems, and a polyanalytic analogue of the fundamental theorem of algebra, plus Radó-type results. Together these form a robust foundation for further hypoanalytic/CR-geometry developments and potential applications in complex analysis and several complex variables. The exposition emphasizes pure mathematics, rigorous proofs, and a spectrum of classical-to-modern techniques, charting trajectories for ongoing advances in polyanalytic function theory.

Abstract

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.

Paper Structure

This paper contains 145 sections, 505 theorems, 2527 equations.

Table of Contents

  1. Basic properties
  2. Defining equations and immediate properties
  3. $q$-analyticity in the sense of Ahern-Bruna
  4. The areolar derivative
  5. The Pompieu operator
  6. $\alpha$-analytic functions on complex manifolds
  7. Classical extension
  8. The real and imaginary parts of $\alpha$-analytic functions
  9. Local properties of classical kind
  10. Zero sets
  11. Some preliminary proof-tools
  12. Schwarz functions
  13. Some lemmas from Nevanlinna theory
  14. Polyanalytic version of the fundamental theorem of algebra
  15. Inner uniqueness
  16. ...and 130 more sections

Key Result

Proposition 1

Let $v$ be an $n+1$-form with $C^k$-smooth coefficients, $k\geq 1$ on an open convex subset $U$ of a finite dimensional vector space $V.$ If $dv=0$ then there exists an $n$-form $u$ on $U$ with $C^k$-smooth coefficients such that $du=v.$

Theorems & Definitions (1140)

  • Proposition 1
  • proof
  • Definition 1.1.1: $q$-analytic functions
  • Definition 1.1.2: Polyanalytic functions of order $q$
  • Definition 1.1.3
  • Definition 1.1.4
  • Definition 1.1.5: Elliptic operator
  • Definition 1.1.6: Hypoelliptic operator
  • Proposition 1.1.7
  • proof
  • ...and 1130 more