A connection between Schur multiplication and Fourier interpolation. II

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 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 ∈   μ‖ 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)