Some H ∞ -Interpolating Sequences and the Behavior of Certain of Their Blaschke Products

Geometric conditions for interpolation

“…Another characterization is due to M. L. Weiss [8]; this result contains the previous result of Wortman. (In his paper [8], Weiss was mainly concerned with sequences that lie on special curves in A that are tangent to 3A at 1.…”

“…By the same hypothesis m(n*) -n* «S A + 1. This establishes the claim and, hence, it follows that (1)(2)(3)(4)(5)(6)(7)(8)(9) 2 \Ln\<(2N)\L\.…”

“…Note that if zm,n) E dG(zn), then one of Rn or Ln equals 0. In any case it is easy to see that | zn -zm( n) | < | RM \ +\ Ln\ , and by ( 1.3) we have (1)(2)(3)(4)(5)(6)(7)(8) A'* 2 M*")< 2 (\R"\+\L"\).…”

“…(In his paper [8], Weiss was mainly concerned with sequences that lie on special curves in A that are tangent to 3A at 1. In his statement of the following theorem, he included the hypothesis that {z"} -» 1 tangentially; however, this condition is readily seen to be extraneous.)…”

Parts in $H\sp{\infty }$ with homeomorphic analytic maps

“…In fact, if h0 £ X and {1 + in(a)} is a net in N converging to Aq, then <Jk(h0) is the limit of the net {1 + i(n(a) + k)}. It is not hard to see from [1] (or [2]) that if P is the part of h0, then X n P consists exactly of the points {ok(h¿): k an integer). Furthermore, in terms of the analytic map, rh , ok(h0) = ta (1 + ik).…”