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Fischer decompositions for entire functions of sufficiently low order

J. M. Aldaz, H. Render

TL;DR

We address Fischer decompositions in the space of all entire functions $E(\\mathbb{C}^d)$, focusing on injectivity of the Fischer operator when the input function has sufficiently low order. Building on existence results for weak Fischer decompositions under Khavinson–Shapiro bounds, the paper proves uniqueness (injectivity) of the Fischer operator under parameter-dependent order constraints, sometimes matching existing bounds and sometimes more restrictive. The proof combines the apolar inner product, a nonnegative-sequence lemma, and precise growth estimates for homogeneous components of entire functions of finite order to show that $P_k^{*}(D)(P\\varphi)=0$ implies $\\varphi=0$ when $\\rho< 2(k-\\beta_2)/(k-\\tau)$ (or the alternate bound when $\\beta_2-\\tau<0$). These results bridge existence and uniqueness regimes and extend Fischer theory to multi-variable, non-homogeneous polynomials.

Abstract

The existence of decompositions of the form $f=P\cdot q+r$ with $P_k^{\ast}\left( D\right) r=0$, where $f$ is entire, $P$ a polynomial and $P^{\ast}_k$ the principal part of $P$ with its coefficients conjugated, was achieved in \cite{AlRe23} under certain restrictions on the order of $f$. Here we prove uniqueness, thereby obtaining Fischer decompositions, under conditions that sometimes match those required for existence, and sometimes are more restrictive, depending on the parameters involved.

Fischer decompositions for entire functions of sufficiently low order

TL;DR

We address Fischer decompositions in the space of all entire functions , focusing on injectivity of the Fischer operator when the input function has sufficiently low order. Building on existence results for weak Fischer decompositions under Khavinson–Shapiro bounds, the paper proves uniqueness (injectivity) of the Fischer operator under parameter-dependent order constraints, sometimes matching existing bounds and sometimes more restrictive. The proof combines the apolar inner product, a nonnegative-sequence lemma, and precise growth estimates for homogeneous components of entire functions of finite order to show that implies when (or the alternate bound when ). These results bridge existence and uniqueness regimes and extend Fischer theory to multi-variable, non-homogeneous polynomials.

Abstract

The existence of decompositions of the form with , where is entire, a polynomial and the principal part of with its coefficients conjugated, was achieved in \cite{AlRe23} under certain restrictions on the order of . Here we prove uniqueness, thereby obtaining Fischer decompositions, under conditions that sometimes match those required for existence, and sometimes are more restrictive, depending on the parameters involved.
Paper Structure (5 sections, 6 theorems, 63 equations)

This paper contains 5 sections, 6 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. A lemma on sequences of nonnegative numbers
  3. The apolar inner product, and statement of the main result
  4. Entire functions of finite order
  5. Proof of Uniqueness

Key Result

Theorem 1

Let $0 \le\beta\leq k-1$, let $P$ be a non-homogeneous polynomial of degree $k \ge 2$ with homogenous expansion $P=P_{0}+\cdots+P_{\beta}+P_{k}$, and let $\varphi:\mathbb{C}^{d}\rightarrow\mathbb{C}$ be an entire function which satisfies $P_{k}^{\ast}\left( D\right) \left( P\varphi\right) =0$ a

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 2
  • proof
  • Theorem 3
  • Lemma 4
  • Proposition 5
  • proof
  • Lemma 6
  • proof