T. Hosokawa, K. Izuchi and D. Zheng recently introduced the concept of asymptotic interpolating sequences (of type 1) in the unit disk for H ∞ (D). It is shown that these sequences coincide with sequences that are interpolating for the algebra QA. Also a characterization is given of the interpolating sequences of type 1 for H ∞ (D), and asymptotic interpolating sequences in the spectrum of H ∞ (D) are studied. The existence of asymptotic interpolating sequences of type 1 for H ∞ (Ω) on arbitrary domains is verified. It is shown that any asymptotic interpolating sequence in a uniform algebra eventually is interpolating.

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We extend a result of M. Heins by showing that for any sequence of points (z n ) in the unit disk D tending to the boundary, there is a Blaschke product B which is universal for noneuclidian translates in the sense that the set {B((z + z n )/(1 + z n z)) : n ∈ N} is locally uniformly dense in the set of all holomorphic functions bounded by one on D. From this, we conclude that for every countable set L of hyperbolic/parabolic automorphisms of the unit disk there exists a Blaschke product which is a common cyclic vector in H 2 for the composition operators associated with the elements in L. These results are obtained by transferring the associated approximation problems to interpolation problems on the corona of H ∞ .

Abstract. We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in R n such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every polynomial in R n can be uniquely written as the sum of a harmonic function and a polynomial multiple of a quadratic function, thus extending a theorem of Ernst Fischer. We then use this decomposition to reduce the Dirichlet problem to a manageable system of linear equations. The algorithm requires differentiation of the boundary function, but no integration. We also show that the polynomial solution produced by our algorithm is the unique polynomial solution, even on unbounded domains such as elliptic cylinders and paraboloids.