Interpolating sequences and extremal problems

In 1958 Carleson proved that a sequence is interpolating if it is uniformly separated, later in 1983 Peter Jones gave a simple constructive proof in [JON] which is stunning in his time. On the other hand, when a sequence is not uniformly separated, Garnett gave a sharp positive result in [GAR2]. Following Jones's idea in [JON] and also a highly relevant paper [VIN] due to Vinogradov, we give a constructive proof to Garnett's result on nonuniformly separated sequence with slightly modified assumption. We will also discuss extremal problems that is described in the language of Hardy spaces and solve it in functional analytical point of view.