In this article, we examine Rational Blaschke functions to create the Multiresolution analysis on the Hardy space \(H_{2} (\mathbb {T})\) . We discuss a decomposition using a non-Blashke sequence, which is analogous to the Whitney cube decomposition of the unit disc. Our primary goal is to successfully recreate an analytic function from samples at the non-Blaschke sequence. We explore the Banach frame structure that was produced from the non-Blaschke sequences, look at the frame structure of the reproducing kernel that corresponds to it, and derive a series representation of any operator in the space in terms of the sampling sequence.
