Near-minimax Interpolation by a Polynomial in z and z-1 on a Circular Annulus

A polynomial of degree n in z^–1 and n–1 in z isdefined by an interpolation projection from the space A(N_p) of functions f analytic in the circular annulusp^–1 < <p and continuous on itsboundaries = p^–1, p. The points ofinterpolation are chosen to be spaced at equal angles aroundthe two boundaries, with arguments on the inner boundary midwaybetween those on the outer boundary. By calculating the Lebesgueconstants numerically, is found to be close to a minimax approximation for all p 1and all degrees n in the range 1 n 15. In the limiting casesp = 1 and, it is proved that is asymptotic to 2^–1 log n. More specifically and , where _nis the Lebesgueconstant of Gronwall for equally spaced interpolation on a circleby a polynomial of degree n. It is also demonstrated that is not in general monotonic in p, and that is not everywhere differentiable in p.