Perturbation of frames for a subspace of a Hilbert space

A frame sequence $[\left\{ {{f_i}} \right\}_{i = 1}^\infty $ in a Hilbert space H allows every element in the closed linear span, [fi], to be written as an infinite linear combination of the frame elements fi. Thus a frame sequence can be considered to be some kind of "generalized basis." Using an extension of a classical condition, we prove that a perturbation $\left\{ {{g_i}} \right\}_{i = 1}^\infty $ of a frame sequence $\left\{ {{f_i}} \right\}_{i = 1}^\infty $ is again a frame sequence whenever the gap from [gi] to [fi] is small enough.

In the special case of a Riesz sequence $\left\{ {{f_i}} \right\}_{i = 1}^\infty $ the gap condition may be omitted.