A Note on Perturbation Stability of Frames and Their Duals

This paper investigates the stability of frame expansions in separable Hilbert spaces under small perturbations of the frame elements. Frames provide redundant representations for vectors, which is essential in many applications such as signal processing and data compression. We establish a perturbation theorem showing that if a frame is perturbed by a bounded operator with sufficiently small norm, the resulting sequence remains a frame with explicitly computable bounds. Furthermore, we demonstrate that the frame expansion coefficients for any vector are stable under such perturbations. An illustrative example is provided to confirm the theoretical results. This work contributes to the theoretical foundations of frame theory and has potential applications in robust data analysis and numerical algorithms.

کلیدواژه ها:

Frame theory ، Hilbert space ، Perturbation ، Stability ، Redundancy ، Frame bounds