Abstract: | Given m × n matrices A = [ajk ] and B = [bjk ], their Schur product is the m × n matrix A ○ B = [ajkbjk ]. For any matrix T, define ‖T‖ S = maxX ≠O ‖T ○ X ‖/‖X ‖ (where ‖·‖ denotes the usual matrix norm). For any complex (2n – 1)‐tuple μ = (μ –n +1, μ –n +2, …, μ n –1), let Tμ be the Hankel matrix [μ –n +j +k –1]j,k and define ??μ = {f ∈ L 1[–π, π] : f? (2j ) = μj for –n + 1 ≤ j ≤ n – 1} . It is known that ‖Tμ‖ S ≤ infequation/tex2gif-inf-18.gif ‖f ‖1. When equality holds, we say Tμ is distinguished. Suppose now that μ j ∈ ? for all j and hence that Tμ is hermitian. Then there is a real n × n hermitian unitary X and a real unit vector y such that 〈(Tμ ○ X )y, y 〉 = ‖Tμ ‖S . We call such a pair a norming pair for Tμ . In this paper, we study norming pairs for real Hankel matrices. Specifically, we characterize the pairs that norm some distinguished Schur multiplier Tμ . We do this by giving necessary and suf.cient conditions for (X, y ) to be a norming pair in the n ‐dimensional case. We then consider the 2‐ and 3‐dimensional cases and obtain further results. These include a new and simpler proof that all real 2 × 2 Hankel matrices are distinguished, and the identi.cation of new classes of 3 × 3 distinguished matrices. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) |