[Paper Review] Polynomial estimates and radius of analyticity on real Banach spaces

Marios K. Papadiamantis|arXiv (Cornell University)|Oct 29, 2012

Advanced Banach Space Theory8 references3 citations

TL;DR

This paper establishes polynomial estimates and bounds the radius of analyticity for power series on real Banach spaces, proving that the radius of analyticity is at least $\rho/\sqrt{2}$, where $\rho$ is the radius of uniform convergence. The results extend classical estimates from complex to real Banach spaces, particularly for $\ell_p$ spaces, using norm comparisons and Stirling-type bounds on multilinear forms.

ABSTRACT

A generalization of Problem 73 of Mazur and Orlicz in the Scottish Book was introduced from L. Harris. The exact value of the constant that appears there is known when complex normed linear spaces are considered. In this paper, we give estimates in the case of an arbitrary real normed linear space and a real $\ell_p$ space. Moreover, if $F(x)$ is a power series, $ρ$ its radius of uniform convergence and $ρ_{\substack{A}}$ its radius of analyticity, we prove that $ρ_{\substack{A}}\geqρ/\sqrt{2}$ and give some respective results for the $n$th Fréchet derivative of $F(x)$.

Motivation & Objective

• To generalize Problem 73 of Mazur and Orlicz from complex to real Banach spaces, particularly for $\ell_p$ spaces.
• To estimate the constant $c(k_1,\ldots,k_n,X)$ controlling the norm of symmetric multilinear forms in terms of their associated homogeneous polynomials.
• To determine lower bounds for the radius of analyticity $\rho_A$ of a power series in terms of its radius of uniform convergence $\rho$.
• To analyze the convergence behavior of Fréchet derivatives of power series in real Banach spaces.

Proposed method

• Derives a sharp upper bound for $|L(x_1^{k_1}\cdots x_n^{k_n})|$ using symmetric multilinear forms and their associated polynomials.
• Applies Stirling's approximation to estimate moments of random variables in the derivation of norm bounds for multilinear forms.
• Uses the inequality $\|L\|_{(n)} \leq \sqrt{n}\|\widehat{L}\|$ to bound the ratio $\|L\|_{(n)}/\|\widehat{L}\|$ for real Banach spaces.
• Applies Lemma 2.1 to relate the growth of coefficients in the Taylor series to the radii of uniform convergence and analyticity.
• Establishes that $\limsup_{m\to\infty} \|L_m\|_{(2)}^{1/m} \leq \sqrt{2}/\rho$ to derive the lower bound $\rho_A \geq \rho/\sqrt{2}$.
• Applies results from Chae on Fréchet derivatives to derive radii of convergence for $D^nF(x)$, showing $\rho_A \geq \rho/\sqrt{2}$ for $n=2$ and $\rho/\sqrt{e}$ for $n \geq 3$.

Experimental results

Research questions

• RQ1 What is the optimal constant $c(k_1,\ldots,k_n,X)$ such that $|L(x_1^{k_1}\cdots x_n^{k_n})| \leq c(k_1,\ldots,k_n,X)\|\widehat{L}\|$ in real Banach spaces?
• RQ2 How does the radius of analyticity $\rho_A$ relate to the radius of uniform convergence $\rho$ for power series on real Banach spaces?
• RQ3 What are the convergence radii of the Fréchet derivatives $D^nF(x)$ in terms of $\rho$?
• RQ4 Can the complex-variable bound $c(k_1,\ldots,k_n,X) = \frac{k_1!\cdots k_n!}{k_1^{k_1}\cdots k_n^{k_n}}\frac{m^m}{m!}$ be extended to real Banach spaces?
• RQ5 What are the sharp estimates for $\|L\|_{(n)}$ in terms of $\|\widehat{L}\|$ in real $\ell_p$ spaces?

Key findings

• The radius of analyticity $\rho_A$ of a power series on a real Banach space satisfies $\rho_A \geq \rho/\sqrt{2}$, where $\rho$ is the radius of uniform convergence.
• For the second Fréchet derivative $DF(x)$, the radius of uniform convergence of its Taylor series is at least $\rho/\sqrt{2}$.
• For $n \geq 3$, the radius of uniform convergence of the Taylor series of $D^nF(x)$ is at least $\rho/\sqrt{e}$.
• The norm ratio $\|L\|_{(2)}/\|\widehat{L}\|$ satisfies $\left(\|L\|_{(2)}/\|\widehat{L}\|\right)^{1/m} \leq \sqrt{2}$ for real Banach spaces.
• For $n \geq 2$, the ratio $\|L\|_{(n)}/\|\widehat{L}\|$ satisfies $\left(\|L\|_{(n)}/\|\widehat{L}\|\right)^{1/m} \leq \sqrt{n}$ in real Banach spaces.
• The bound $\|L\|_{(n)} \leq \sqrt{n}\|\widehat{L}\|$ is sharp in the sense that equality is approached when $m$ is even and $k_1 = k_2 = m/2$.