What is "overcompleteness"

n. The condition of being overcomplete

A subset {ϕ} of a Banach space X, sometimes called a "system", is complete if every element in X can be approximated arbitrarily well in norm by finite linear combinations of elements in {ϕ}. Such a complete system is overcomplete if removal of a ϕ from the system results in a system (i.e., {ϕ}) that is still complete. In different research, such as signal processing and function approximation, overcompleteness can help researchers to achieve a more stable, more robust, or more compact decomposition than using a basis. Overcomplete frames are widely used in mathematics, computer science, engineering, and statistics.