Abstract
Given m × n matrices A = [a~jk~ ] and B = [b~jk~ ], their Schur product is the m × n matrix A ○ B = [a~jk~b~jk~ ]. For any matrix T, define ‖T‖ ~S~ = max~X ≠O~ ‖T ○ X ‖/‖X ‖ (where ‖·‖ denotes the usual matrix norm). For any complex (2__n__ – 1)‐tuple μ = (μ ~–n +1~, μ ~–n +2~, …, μ ~n –1~), let T~μ~ be the Hankel matrix [μ –~n +j +k –1~]~j,k~ and define
𝔅~μ~ = {f ∈ L ~1~[–π, π] : f̂ (2__j__ ) = μ~j~ for –n + 1 ≤ j ≤ n – 1} .
It is known that ‖T~μ~‖ ~S~ ≤ inf ‖f ‖~1~. When equality holds, we say T~μ~ is distinguished. Suppose now that μ ~j~ ∈ ℝ for all j and hence that T~μ~ is hermitian. Then there is a real n × n hermitian unitary X and a real unit vector y such that 〈(T~μ~ ○ X )y, y 〉 = ‖T~μ~ ‖~S~ . We call such a pair a norming pair for T~μ~ . In this paper, we study norming pairs for real Hankel matrices. Specifically, we characterize the pairs that norm some distinguished Schur multiplier T~μ~ . 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)