Overcompleteness

In linear algebra, a subset of the vectors ${\displaystyle \{\phi _{i}\}_{i\in J}}$ of a Banach space ${\displaystyle X}$, sometimes called a "system", is complete if every element in ${\displaystyle X}$ can be approximated arbitrarily well in norm by finite linear combinations of elements in ${\displaystyle \{\phi _{i}\}_{i\in J}}$. Such a complete system is overcomplete if removal of a ${\displaystyle \phi _{j}}$ from the system results in a system (i.e., ${\displaystyle \{\phi _{i}\}_{i\in J\backslash \{j\}}}$) 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.

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