Perturbation of Frames

Abstract

In applications where bases appear, the question of stability plays an important role. That is, if {f _k } ^∞ _k=1 is a basis and {g _k } ^∞ _k=1 is in some sense “close” to {f _k } ^∞ _k=1 does it follows that {g _k } ^∞ _k=1 is also a basis? A classical result states that if {f _k } ^∞ _k=1 is a basis for a Banach space X, then a sequence {g _k } ^∞ _k=1 in X is also a basis if there exists a constant λ ∈]0,1[ such that for all finite sequences of scalars {c _k }. The result is usually attributed to Paley and Wiener [231], but it can be traced back to Neumann [226]: in fact, it is an almost immediate consequence of Theorem A.5.3 with Uf _k := g _k .