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A Fischer type decomposition theorem from the apolar inner product

J. M. Aldaz, H. Render

Abstract

We continue the study initiated by H. S. Shapiro on Fischer decompositions of entire functions, showing that such decomposition exist in a weak sense (we do not prove uniqueness) under hypotheses regarding the order of the entire function $f$ to be expressed as $f= P\cdot q+r$, the polynomial $P$, and bounds on the apolar norm of homogeneous polynomials of degree $m$. These bounds, previously used by Khavinson and Shapiro, and by Ebenfelt and Shapiro, can be interpreted as a quantitative, asymptotic strengthening of Bombieri's inequality. In the special case where both the dimension of the space and the degree of $P$ are two, we characterize for which polynomials $P$ such bounds hold.

A Fischer type decomposition theorem from the apolar inner product

Abstract

We continue the study initiated by H. S. Shapiro on Fischer decompositions of entire functions, showing that such decomposition exist in a weak sense (we do not prove uniqueness) under hypotheses regarding the order of the entire function to be expressed as , the polynomial , and bounds on the apolar norm of homogeneous polynomials of degree . These bounds, previously used by Khavinson and Shapiro, and by Ebenfelt and Shapiro, can be interpreted as a quantitative, asymptotic strengthening of Bombieri's inequality. In the special case where both the dimension of the space and the degree of are two, we characterize for which polynomials such bounds hold.
Paper Structure (7 sections, 15 theorems, 179 equations)

This paper contains 7 sections, 15 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Basic properties of the apolar inner product
  3. Apolar norms of products of linear factors
  4. Observations on amenability and the bounds of Khavinson and Shapiro
  5. Special cases: when $k= 1$ or $d=1$ the conjecture is true
  6. Proof of Theorem \ref{['main']}
  7. Fischer decompositions on certain Banach spaces of entire functions

Key Result

Theorem 2

Let $P_{k}$ be a homogeneous polynomial of degree $k > 0$ on $\mathbb{C}^d$, and let us write $T:=T_{P_{k}}$, where $T_{P_{k}}$ is defined by (linealT). Assume that there exist a $C=C(P_{k})>0$ ($C$ independent of $m$) and an $\tau\in \{0, \dots , k\}$, such that for every $m>0$ and every homogeneou If for $0\leq j<k$ the polynomials $P_{j}\left( z\right)$ are homogeneous of degree $j$, and for

Theorems & Definitions (29)

  • Definition 1
  • Theorem 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • proof
  • Example 6
  • Theorem 7
  • ...and 19 more