Let there be given a function
f(z) analytic in an open connected set, not necessarily simply connected,
which is bounded by simple closed analytic curves such that the function is
continuous on the closure of the region and such that the real part of the
function satisfies boundary conditions that are analytic in a neighborhood of
the boundary. We want to interpolate f(z) along the boundaries and find
conditions that make the interpolants converge maximally to f(z) throughout
the closure of the region. The boundary condition on the real part of f(z)
permits the analytic continuation of f(z) across the boundary curves and
ensures that we are interpolating at points interior to the region of analyticity.
In our error estimates (Theorem 1) maximal convergence depends in an
essential way on how far we can reflect f(z) and this in turn depends on the
boundary values of the real part of f(z) as well as on the geometry of the
given region and its analytic boundaries. In Theorems 2 and 3, a simply
connected region is considered. Special points of interpolation are given, these
depend only on the parametric representation of the boundary curves and
not a conformal map. These points are the image points of the Chebyshev
polynomials.
