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_jkb_jk ]. For any matrix T, define ‖T‖ _S = max_{X ≠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 T_μ be the Hankel matrix μ –_{n +j +k –1}]_j,k and define ??_μ = {f ∈ L ₁–π, π] : f? (2j ) = μ_j for –n + 1 ≤ j ≤ n – 1} . It is known that ‖T_μ‖ _S ≤ infequation/tex2gif-inf-18.gif ‖f ‖₁. When equality holds, we say T_μ is distinguished. Suppose now that μ _j ∈ ? 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)