Localized Bounded Below Approximate Schauder Frames are Finite Unions of Approximate Riesz Sequences

Based on the truth of Feichtinger conjecture by Marcus, Spielman and Srivastava $\mathit{\text{[Ann.}}\mathit{\text{of}}\mathit{\text{Math.}}\mathit{\text{(2),}}\mathit{\text{2015]}}$ and from the localized version by Gröchenig $\mathit{\text{[Adv.}}\mathit{\text{Comput.}}\mathit{\text{Math.,}}\mathit{\text{2003]}}$, we introduce the notion of localization of approximate Schauder frames (ASFs) and approximate Riesz sequences (ARSs). We show that localized bounded below ASFs are finite unions of ARSs.