Overcompleteness
Overcompleteness is a concept from linear algebra that is widely used in mathematics, computer science, engineering, and statistics (usually in the form of overcomplete frames). It was introduced by R. J. Duffin and A. C. Schaeffer in 1952. Formally, a subset of the vectors of a Banach space , sometimes called a "system", is complete if every element in can be approximated arbitrarily well in norm by finite linear combinations of elements in . A complete system is additionally overcomplete if there exists a which can be removed from the system while maintaining completeness (i.e., ). In this sense, the system contains more vectors than necessary to be complete, hence "overcomplete". In research areas 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. Relation between overcompleteness and frames Overcompleteness is usually discussed as a property of overcomplete frames. The theory of frame originates in a paper by Duffin and Schaeffer on non-harmonic Fourier series. The frame is defined to be a set of non-zero vectors such that for an arbitrary , where denotes the inner product, and are positive constants called bounds of