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Homogeneous polynomials and extensions of Hardy‐Hilbert's inequality
Author(s) -
Anagnostopoulos Vasileios A.,
Sarantopoulos Yannis,
Tonge Andrew M.
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201010035
Subject(s) - mathematics , hermitian matrix , homogeneous , hilbert space , trigonometry , norm (philosophy) , polynomial , homogeneous polynomial , combinatorics , pure mathematics , algebra over a field , mathematical analysis , matrix polynomial , political science , law
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.