Overcompleteness

A subset $\{\phi_i\}_{i\in J}$ 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 $\{\phi_i\}_{i\in J}$ . Such a complete system is overcomplete if removal of a $\phi_j$ from the system results in a system (i.e., $\{\phi_i\}_{i\in J\backslash\{j\}}$ ) 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.

单词	Overcomplete
释义	Overcomplete 英语百科 Overcompleteness A subset $\{\phi_i\}_{i\in J}$ 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 $\{\phi_i\}_{i\in J}$ . Such a complete system is overcomplete if removal of a $\phi_j$ from the system results in a system (i.e., $\{\phi_i\}_{i\in J\backslash\{j\}}$ ) 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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