We show that there exists a closed infinite dimensional subspace of ${{H}^{\infty }}\left( {{B}^{n}} \right)$ such that every function of norm one is universal for some sequence of automorphisms of ${{B}^{n}}$.

A Blaschke product $B$ with zero-sequence $(a_n)$ is called almost interpolating if the inequality $\liminf_n(1-|a_n|^2)|B'(a_n)|\geqslant \d>0$ holds. The sets $U$ for which there exists a Blaschke product $B$ such that $(a-B)/(1-\ov a B)$ is almost interpolating if and only if $a \in U$ are studied. Examples of such sets include open sets, containing the origin, and whose complement is the closure of an arbitrary set of concentric open arcs around the origin or open sets whose complement is of zero logarithmic capacity. Results on the range of interpolating Blaschke product s on the set of trivial points in the spectrum of $H^{\infty}$ are deduced.

We show that there exists a singular inner function $S$ which is universal for noneuclidean translates; that is one for which the set $\{S(\frac{z\,+\,{{z}_{n}}}{1\,+\,{{{\bar{z}}}_{n}}z})\,:\,n\,\in \,\mathbb{N}\}$ is locally uniformly dense in the set of all zero-free holomorphic functions in $\mathbb{D}$ bounded by one.

We extend a result of M. Heins by showing that for any sequence of points $(z_n)$ in the unit disk ${\Bbb D}$ tending to the boundary, there is a Blaschke product $B$ which is universal for noneuclidian translates in the sense that the set $\{B((z\,{+}\,z_n)/(1\,{+}\,\overline{z}_nz))\,{:} n\,{\in}\,{\Bbb N}\}$ is locally uniformly dense in the set of all holomorphic functions bounded by one on ${\Bbb D}$. From this, we conclude that for every countable set ${\sp L}$ 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 ${\sp L}$. These results are obtained by transferring the associated approximation problems to interpolation problems on the corona of $H^\infty$.

T. Hosokawa, K. Izuchi and D. Zheng recently introduced the concept of asymptotic interpolating sequences (of type 1) in the unit disk for $H^\infty$(${\bb D}$). 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 $H^\infty$(${\bb D}$), and asymptotic interpolating sequences in the spectrum of $H^\infty$(${\bb D}$) are studied. The existence of asymptotic interpolating sequences of type $1$ for $H^\infty(\Omega)$ on arbitrary domains is verified. It is shown that any asymptotic interpolating sequence in a uniform algebra eventually is interpolating.

Let A be a Banach algebra and let B be a linear subspace of A. Recall that A has the Dunford Pettis property if whenever ƒn→ 0 weakly in A* and φn → 0 weakly in A* then φn(ƒn) → 0. Bourgain showed that H∞ has the Dunford Pettis property using the theory of ultraproducts. The Dunford Pettis property is related to the notion of Bourgain algebra, denoted Bb, introduced by [6] Cima and Timoney. The algebra Bb is the set of ƒ in A such that if ƒn → 0 weakly in B then dist(ƒƒn, B) —> 0. Bourgain showed [2] that a closed subspace X of C(L)y where L is a compact Hausdorff space, has the Dunford Pettis property if Xb — C(L). Cima and Timoney proved that Bb is a closed subalgebra of A and that if B is an algebra then B⊂Bb. In this paper we study the Bourgain algebra associated with various algebras of functions on the unit circle T.