Polarization constant $\mathcal{K}(n,X)=1$ for entire functions of exponential type

Pappas, A.; Papadopoulos, P.; Athanasopoulou, L.

doi:10.22075/ijnaa.2015.252

Polarization constant $\mathcal{K}(n,X)=1$ for entire functions of exponential type

Document Type : Research Paper

Authors

A. Pappas ¹

P. Papadopoulos ²

L. Athanasopoulou ³

¹ Civil Engineering Department, School of Technological Applications, Piraeus University of Applied Sciences (Technological Education Institute of Piraeus), GR 11244, Egaleo, Athens, Greece

² adepartment of electronics engineering, school of technological applications, technological educational institution (tei) of piraeus, gr 11244, egaleo, athens, Greece.

³ Department of Electronics Engineering, School of Technological Applications, Piraeus University of Applied Sciences (Technological Education Institute of Piraeus), GR 11244, Egaleo, Athens, Greece

10.22075/ijnaa.2015.252

Abstract

In this paper we will prove that if $L$ is a continuous symmetric n-linear form on a Hilbert space and $\widehat{L}$ is the associated continuous n-homogeneous polynomial, then $||L||=||\widehat{L}||$. For the proof we are using a classical generalized inequality due to S. Bernstein for entire functions of exponential type. Furthermore we study the case that if X is a Banach space then we have that
$$
|L|=|\widehat{L}|, \forall L \in{\mathcal{L}}^{s}(^{n}X).
$$
If the previous relation holds for every $L \in {\mathcal{L}}^{s}\left(^{n}X\right)$, then spaces ${\mathcal{P}}\left(^{n}X\right)$ and $L \in {\mathcal{L}}^{s}(^{n}X)$ are isometric. We can also study the same problem using Fr$\acute{e}$chet derivative.

Keywords

Polarization constants

polynomials on Banach spaces

polarization formulas