Matolcsi, Máté and Munoz, G. A. (2006) On the real linear polarization constant problem. MATHEMATICAL INEQUALITIES & APPLICATIONS, 9 (3). pp. 485494. ISSN 13314343

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Abstract
The present paper deals with lower bounds for the norm of products of linear forms. It has been proved by J. AriasdeReyna [2], that the socalled n(th) linear polarization constant c(n)(Cn) is n(n/2), for arbitrary n is an element of N. The same value for c(n) (Rn) is only conjectured. In a recent work A. Pappas and S. Revesz prove that c(n) (Rn) = n(n/2) for n <= 5. Moreover, they show that if the linear forms are given as f(j)(x) = [x, a(j)),for some unit vectors a(j) (1 <= j <= n), then the product of the f(j)'s attains at least the value n(n/2) at the normalized signed sum of the vectors having maximal length. Thus they asked whether this phenomenon remains true for arbitrary n is an element of N. We show that for vector systems {a(j)}(j=1)(n) close to an orthonormal system, the PappasRevesz estimate does hold true. Furthermore, among these vector systems the only system giving n(n/2) as the norm of the product is the orthonormal system. On the other hand, for arbitrary vector systems we answer the question of A. Pappas and S. Revesz in the negative when n is an element of N is large enough. We also discuss various further examples and counterexamples that may be instructive for further research towards the determination of c(n)(Rn).
Item Type:  Article 

Uncontrolled Keywords:  Polynomials; Polynomial norm estimates; Linear polarizations constants; NORMS; SPACES; LOWER BOUNDS; Plank problem 
Subjects:  Q Science / természettudomány > QA Mathematics / matematika 
SWORD Depositor:  MTMT SWORD 
Depositing User:  MTMT SWORD 
Date Deposited:  10 Dec 2013 13:40 
Last Modified:  10 Dec 2013 13:40 
URI:  http://real.mtak.hu/id/eprint/7946 
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