On discrete norms of polynomials (original) (raw)
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Mathematical proceedings of the Cambridge Philosophical Society, 1998
In this paper we give a different interpretation of Bombieri's norm. This new point of view allows us to work on a problem posed by Beauzamy about the behavior of the sequence S n (P) = sup Q n [P Q n ] 2 , where P is a fixed m−homogeneous polynomial and Q n runs over the unit ball of the Hilbert space of n−homogeneous polynomials. We also study the factor problem for homogeneous polynomials defined on C N and we obtain sharp inequalities whenever the number of factors is no greater than N. In particular, we prove that for the product of homogeneous polynomials on infinite dimensional complex Hilbert spaces our inequality is sharp. Finally, we use these ideas to prove that any set {z k } n k=1 of unit vectors in a complex Hilbert space for which sup z =1 | z, z 1 • • • z, z n | is minimum must be an orthonormal system.
Lower Bounds for Norms of Products of Polynomials
Mathematical Proceedings of the Cambridge Philosophical Society, 1998
In this paper we give a different interpretation of Bombieri's norm. This new point of view allows us to work on a problem posed by Beauzamy about the behavior of the sequence S n (P) = sup Qn [P Q n ] 2 , where P is a fixed m−homogeneous polynomial and Q n runs over the unit ball of the Hilbert space of n−homogeneous polynomials. We also study the factor problem for homogeneous polynomials defined on C N and we obtain sharp inequalities whenever the number of factors is no greater than N. In particular, we prove that for the product of homogeneous polynomials on infinite dimensional complex Hilbert spaces our inequality is sharp. Finally, we use these ideas to prove that any set {z k } n k=1 of unit vectors in a complex Hilbert space for which sup z =1 | z, z 1 • • • z, z n | is minimum must be an orthonormal system.
Inequalities Concerning theLp-Norm of a Polynomial
Journal of Mathematical Analysis and Applications, 1998
In this paper we obtain L p , p G 1, inequalities for the class of polynomials < < having no zeros in z-K, K G 1. Our result generalizes as well as improves upon some well known results.
Inequalities Concerning Maximum Modulus of Polynomials
Thai Journal of Mathematics, 2012
If p(z) = a 0 + n j=µ a j z j , 1 ≤ µ ≤ n is a polynomial of degree n having no zeros in |z| < k, k > 0, then for 0 ≤ r ≤ ρ ≤ k, [A. Aziz, W.M. Shah, Inequalities for a polynomial and its derivative, Math. Inequal. and Appl. 7 (3) (2004) 397-391], M (p ′ , ρ) ≤ nρ µ−1 (ρ µ + k µ) n µ −1 (k µ + r µ) n µ {M (p, r) − m(p, k)}. In this paper, we have generalized as well as improved upon the above inequality by involving the coefficients of the polynomial p(z). Besides, our result gives interesting refinements of some well-known results.
The sup‐norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials
Mathematische Nachrichten, 2019
Let A m p,r (n) be the best constant that fulfills the following inequality: for every mhomogeneous polynomial P (z) = |α|=m a α z α in n complex variables, |α|=m |a α | r 1/r ≤ A m p,r (n) sup z∈B ℓ n p P (z). For every degree m, and a wide range of values of p, r ∈ [1, ∞] (including any r in the case p ∈ [1, 2], and any r and p for the 2-homogeneous case), we give the correct asymptotic behavior of these constants as n (the number of variables) tends to infinity. Remarkably, in many cases, extremal polynomials for these inequalities are not (as traditionally expected) found using classical random unimodular polynomials, and special combinatorial configurations of monomials are needed. Namely, we show that Steiner polynomials (i.e., m-homogeneous polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems), do the work for certain range of values of p, r. As a byproduct, we present some applications of these estimates to the interpolation of tensor products of Banach spaces, to the study of (mixed) unconditionality in spaces of polynomials and to the multivariable von Neumann's inequality.
Publications of the Research Institute for Mathematical Sciences, 2003
We show that for every Banach space X the set of 2-homogeneous continuous polynomials whose canonical extension to X * * attain their norm is a dense subset of the space of all 2-homogeneous continuous polynomials P(2 X).
Where do homogeneous polynomials on ¿ 1 n attain their norm?
Journal of Approximation Theory, 2004
Using a 'reasonable' measure in Pð 2 c n 1 Þ; the space of 2-homogeneous polynomials on c n 1 ; we show the existence of a set of positive (and independent of n) measure of polynomials which do not attain their norm at the vertices of the unit ball of c n 1 : Next we prove that, when n grows, almost every polynomial attains its norm in a face of 'low' dimension.