discreteframe

Discreteframe is a term used in mathematical signal processing and functional analysis to denote a discrete frame: a countable set of vectors in a Hilbert space that allows stable, redundant representation of any element of the space.

Formally, a family {f_k} in a Hilbert space H is a discreteframe if there exist constants A,B > 0 such that for every f in H, A ||f||^2 ≤ sum_k |<f, f_k>|^2 ≤ B ||f||^2. The index set k typically runs over a finite or countably infinite set. Discreteframes generalize orthonormal bases by permitting redundancy while preserving stable reconstruction.

Associated operators provide the framework for analysis and synthesis. The analysis operator T maps f to the sequence {<f, f_k>} in l^2(K), and the synthesis operator T* reconstructs from coefficients via sum_k c_k f_k. The frame operator S = T* T is bounded, positive, and invertible, and every f in H can be reconstructed through f = sum_k <f, f_k> S^{-1} f_k, where {S^{-1} f_k} forms a dual frame. When S is a scalar multiple of the identity, the frame is called tight, and if that multiple is 1, it is a Parseval frame. Orthonormal bases are special cases of Parseval frames.

Discreteframes have diverse constructions and applications. Random frames, Gabor frames, and wavelet frames are well-known examples in finite- and infinite-dimensional settings. They are used in signal and image processing, data compression, numerical linear algebra, and robust representations where redundancy improves stability against noise and erasures.