Non-symmetric polarization

2016/03/14 by Andreas Defant, Defant, Andreas, Sunke Schlüters +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1603.04279

arxiv created 2016/03/14 · arxiv updated 2016/03/15

Abstract

Let P be an m-homogeneous polynomial in n-complex variables x₁, \dotsc, x_n. Clearly, P has a unique representation in the form P(x)= ∑_{1 ≤ j1 ≤ \dotsc ≤ jm ≤ n} c_{(j1, \dotsc, jm)} x_j1 \dotsb x_jm , and the m"~form L_P(x⁽¹⁾, \dotsc, x^(m))= ∑_{1 ≤ j1 ≤ \dotsc ≤ jm ≤ n} c_{(j1, \dotsc, jm)} x⁽¹⁾_j1 \dotsb x^(m)_jm satisfies L_P(x,\dotsc, x) = P(x) for every x∈ℂⁿ. We show that, although L_P in general is non-symmetric, for a large class of reasonable norms ‖ ⋅ ‖ on ℂⁿ the norm of L_P on (ℂⁿ, ‖ ⋅ ‖ )^m up to a logarithmic term (c log n)^m2 can be estimated by the norm of P on (ℂⁿ, ‖ ⋅ ‖ ); here c ≥ 1 denotes a universal constant. Moreover, for the ℓ_p"~norms ‖ ⋅ ‖_p, 1 ≤ p < 2 the logarithmic term in the number n of variables is even superfluous.

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