On the real linear polarization constant problem
نویسندگان
چکیده
منابع مشابه
On the Real Linear Polarization Constant Problem
The present paper deals with lower bounds for the norm of products of linear forms. It has been proved by J. Arias-de-Reyna [2], that the so-called n linear polarization constant cn(C ) is n, for arbitrary n ∈ N. The same value for cn(R ) is only conjectured. In a recent work A. Pappas and S. Révész prove that cn(R ) = n for n ≤ 5. Moreover, they show that if the linear forms are given as fj(x)...
متن کاملOn the Linear Polarization Constant Of
The present work contributes to the determination of the n-th linear polarization constant cn(H) of an n-dimensional real Hilbert space H . We provide some new lower bounds on the value of sup‖y‖=1 | 〈x1, y〉 · · · 〈xn, y〉 |, where x1, . . . , xn are unit vectors in H . In particular, the results improve an earlier estimate of Marcus. However, the intriguing conjecture cn(H) = n n/2 remains open...
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چکیده ندارد.
15 صفحه اولThe dth linear polarization constant of Rd
In [C. Benítez, Y. Sarantopoulos, A. Tonge, Lower bounds for norms of products of polynomials, Math. Proc. Cambridge Philos. Soc. 124 (3) (1998) 395–408] it was conjectured that for all unit vectors u1, . . . , ud in Rd , X (u1, . . . , ud ) := sup x∈Rd , |x|2=d d ∏ i=1 〈x,ui〉 1 with equality occurring iff u1, . . . , ud are orthonormal. We relate this to a conjecture about solutions of Ay = y−...
متن کاملThe d-th linear polarization constant of R
In [4] it was conjectured that for all unit vectors u1, · · · , ud in Rd, X (u1, · · · , ud) := sup x∈Rd, |x|2=d d ∏ i=1 〈x, ui〉 ≥ 1 with equality occurring iff u1, · · · , ud are orthonormal. We relate this to a conjecture about solutions of Ay = y−1, where A = (〈ui, uj〉), and show that if the conjecture fails then the u1, · · · , ud minimizing X must be linearly dependent. We also show X (u1,...
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ژورنال
عنوان ژورنال: Mathematical Inequalities & Applications
سال: 2006
ISSN: 1331-4343
DOI: 10.7153/mia-09-47