Norms of analytic interpolation projections on general domains

Abstract

For any finitely connected open domain D bounded by Jordan curves, it is proved that there exists a function in A( D), the space of functions analytic in D and continuous on D , having a maximum modulus of at most unity and taking given values of modulus at most unity at a finite number of given points on ∂D. Explicit constructions of such functions are given for the special cases of a disc, a domain between two circles, an elliptical ring, and an ellipse with a slit between its foci. It is hence proved that the projection I n from A( D) onto the subspace of polynomials of fixed degree n − 1 which interpolate in n given points of ∂D, has a Chebyshev norm equal to the supremum of the sum of the absolute values of the fundamental polynomials in the Lagrange interpolation formula. Similar results are also proved in the special cases of a domain between two circles with centres ζ 0, ζ 1 and an ellipse with a slit between its foci ±1, when the interpolation subspaces are more appropriately chosen to be, respectively, polynomials of degrees n in ( z − ζ 1) 1 and m − 1 in z − ζ 0 and functions of the form A n ( z)+ √( z 2 1) B n−1 ( z), where A n and B n − 1 are polynomials of respective degrees n and n − 1.