Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Entropy numbers and box dimension of polynomials and holomorphic functions

Daniel Carando, Carlos D'Andrea, Leodan A. Torres, Pablo Turco

Abstract

We study entropy numbers and box dimension of (the image of) homogeneous polynomials and holomorphic functions between Banach spaces. First, we see that entropy numbers and box dimensions of subsets of Banach spaces are related. We show that the box dimension of the image of a ball under a homogeneous polynomial is finite if and only if it spans a finite-dimensional subspace, but this is not true for holomorphic functions. Furthermore, we relate the entropy numbers of a holomorphic function to those of the polynomials of its Taylor series expansion. As a consequence, if the box dimension of the image of a ball by a holomorphic function $f$ is finite, then the entropy numbers of the polynomials in the Taylor series expansion of $f$ at any point of the ball belong to $\ell_p$ for every $p>1$.

Entropy numbers and box dimension of polynomials and holomorphic functions

Abstract

We study entropy numbers and box dimension of (the image of) homogeneous polynomials and holomorphic functions between Banach spaces. First, we see that entropy numbers and box dimensions of subsets of Banach spaces are related. We show that the box dimension of the image of a ball under a homogeneous polynomial is finite if and only if it spans a finite-dimensional subspace, but this is not true for holomorphic functions. Furthermore, we relate the entropy numbers of a holomorphic function to those of the polynomials of its Taylor series expansion. As a consequence, if the box dimension of the image of a ball by a holomorphic function is finite, then the entropy numbers of the polynomials in the Taylor series expansion of at any point of the ball belong to for every .
Paper Structure (5 sections, 8 theorems, 54 equations)

This paper contains 5 sections, 8 theorems, 54 equations.

Table of Contents

  1. Entropy numbers, covering numbers and box dimension
  2. On the dimension of the image
  3. The proof of Theorem \ref{['thm: BoxdimensionPoly']}
  4. On the rank of the jacobian of homogeneous polynomials
  5. Entropy numbers for holomorphic functions and their Taylor coefficients

Key Result

Proposition 1.1

Let $(X,d)$ a metric space and $L\subset X$ a bounded set. If $(\delta_n)_n$ is a decreasing sequence of real numbers such that $\lim\limits_{n\to \infty} \delta_n=0$ and $\liminf\limits_{n\to \infty} \dfrac{\delta_{n+1}}{\delta_n}>0$, then

Theorems & Definitions (25)

  • Proposition 1.1
  • Proposition 1.2
  • proof
  • Lemma 1.3
  • proof
  • Example 1.4
  • proof
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • ...and 15 more