INTERPOLATION OF l" SEQUENCES BY H* FUNCTIONS

It is pointed out that the method used by L. Carleson to study interpolation by bounded analytic functions applies to the study of the analogous problem for H" functions. In particular, there exist sequences of points in the unit disc which are not uniformly separated, but which are such that every l" sequence can be interpolated along this sequence by an Hv function (l^p^q^ + oo). Let (77", /") denote the set of sequences {z„}"=1 in the open unit disc such that {{/(zJ}:/e77^K It was proved by L. Carleson [1] and also by H. S. Shapiro and A. L. Shields [7] that (zj e (7/°°, P°) if and only if {zn} is uniformly separated; that is, n zm — zh Zb-Zr, = ô > 0, « = 1,2, Since 77e0 c77" for all/?<co, (Hp,l'°) contains the uniformly separated sequences. A. K. Snyder [8] has proved the existence of sequences {z„} e (772, /°°) which are not uniformly separated, and P. L. Duren and H. S. Shapiro [4] have given for 0</?< co an explicit construction of sequences {zn} which belong to (77p, /°°) but which are not uniformly separated. We show here that the method used by Carleson [1], as modified by Hörmander [5] and extended by Duren [2], also applies directly to the problem of characterizing (77", l") when \^p, ?=oo. In particular, this method allows one to construct easily sequences in (77^, /°°) which are not uniformly separated. Carleson's method is to convert the interpolation problem to one of estimating a measure associated with the sequence {z„}. The equivalence is given by the following theorem whose proof is a well-known consequence of the Lv—Lv' duality. See, for example [3, Chapter 8]. Received by the editors January 8, 1971 and, in revised form, October 4, 1971. AMS 1970 subject classifications. Primary 30A78, 30A80.