INTERPOLATION MANIFOLDS

A submanifold M of the boundary of a strictly pseudoconvex domain D is called an interpolation manifold for D if at each point p in M, the tangent space to M at p is contained in the maximal complex tangent space of the boundary of D at p. For the unit polydisc U('N) a submanifold M of its distinguished boundary ('N) is an interpolation manifold if at each point p in M, the tangent space to M at p intersects the closed positive cone of the tangent space to ('N) at p only at the origin.; We prove that any compact subset of a sufficiently smooth interpolation manifold for U('N) us a peak-interpolation set for A('0)(U('N)) and an interpolation set and local peak set for A('(INFIN))(U('N)). (We donate by A('k)(U('N)) (0 (LESSTHEQ) k (LESSTHEQ) (INFIN)) the algebra of functions analytic on U('N) which, together with all their derivatives of order less than or equal to k, extend continuously to the closure of U('N).) The peak-interpolation result for A('0)(U('N)) in the case of a one-dimensional interpolation manifold follows also from an interpolation result for sets of zero S-width due to Forelli. We show that Forelli's result does not in general yield our result for higher dimensional manifolds.; Interpolation manifolds for strictly pseudoconvex domains or for the unit polydisc are examples of sets for which we prove a general peak-interpolation result for L('(INFIN))-functions by bounded analytic functions. In the case of strictly pseudoconvex domains or the unit polydisc, they are the only manifolds for which this peak-interpolation result for L('(INFIN))-functions holds.; We prove an interpolation result for C('k)-functions on compact subsets of interpolation manifolds both in the strictly pseudoconvex case and the polydisc case. Using Fourier analysis methods we show by examples that these results are, however, not the best possible and that certain expected kinds of differentiable peak-interpolation results on the ball do not exist.