Homogeneous polynomials and extensions of Hardy-Hilbert's inequality

✍ Scribed by Vasileios A. Anagnostopoulos; Yannis Sarantopoulos; Andrew M. Tonge

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 131 KB
• Volume: 285
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.201010035

✦ Synopsis

Abstract

If L is a continuous symmetric n‐linear form on a real or complex Hilbert space and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\widehat{L}$\end{document} is the associated continuous n‐homogeneous polynomial, then \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Vert L\Vert =\big \Vert \widehat{L}\big \Vert$\end{document}. We give a simple proof of this well‐known result, which works for both real and complex Hilbert spaces, by using a classical inequality due to S. Bernstein for trigonometric polynomials. As an application, an open problem for the optimal lower bound of the norm of a homogeneous polynomial, which is a product of linear forms, is related to the so‐called permanent function of an n × n positive definite Hermitian matrix. We have also derived generalizations of Hardy‐Hilbert's inequality.