Overcompleteness Explained
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
.
[1] A system is called
overcomplete if it contains more vectors than necessary to be complete, i.e., there exist
that can be removed from the system such that
\{\phii\}i\in\setminus\{\phij\}
remains complete. 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.
[2] Relation between overcompleteness and frames
The theory of frames originates in a paper by Duffin and Schaeffer on non-harmonic Fourier series.[3]
Notes and References
- C. Heil, A Basis Theory Primer: Expanded Edition. Boston, MA: Birkhauser, 2010.
- R. Balan, P. Casazza, C. Heil, and Z. Landau, Density, overcompleteness, and localization of frames. I. theory, Journal of Fourier Analysis and Applications, vol. 12, no. 2, 2006.
- R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Transactions of the American Mathematical Society, vol. 72, no. 2, pp. 341