Norms of analytic interpolation projections on general domains
For any finitely connected open domain D bounded by Jordan curves, it is here proved that the projection from A(D), the space of functions analytic in D and continuous on D, onto the subspace of polynomials of fixed degree n-1 which interpolate in given distinct points of \(\partial D\), has a Chebyshev norm equal to the supremum of the sum of the absolute values of the fundamental polynomials in the Lagrange interpolation formula. The proof is based on the construction of a function in A(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 of \(\partial D\). Similar results are also proved in the special cases of a domain between two circles with centres \(z_ 0\) and \(z_ 1\), and an ellipse with a slit between its foci \(\pm 1\), when the interpolation subspaces are more appropriately chosen to be, respectively, polynomials of degree n in \((z- z_ 1)^{-1}\) and m-1 in \(z-z_ 0\), and functions of the form \(A_ n(z)+\sqrt{(z^{2-1})}B_{n-1}(z)\), where \(A_ n\) and \(B_{n-1}\) are polynomials of respective degrees n and n-1. The latter results have applications in fracture mechanics, where such approximations are usual for representing stress functions in a region which includes a crack along [-1,1].
scientific article; zbMATH DE number 3851651