are not unique because numbers less than A and greater than B are also valid frame bounds. The optimal lower bound is the supremum of all lower bounds and the optimal upper bound is the infimum of all upper bounds. A frame is called overcomplete (or redundant) if it is not a basis for the vector space. Analysis operator The operator mapping to a sequence of coefficients is called the analysis operator of the frame. It is defined by: By using this definition we may rewrite the frame condition as where the left and right norms denote the norm in and the middle norm is the norm. Synthesis operator The adjoint operator of the analysis operator is called the synthesis operator of the frame. Motivation for the lower frame bound We want that any vector can be reconstructed from the coefficients . This is satisfied if there exists a constant such that for all we have: By setting and applying the linearity of the analysis operator we get that this condition is equivalent to: for all which is exactly the lower frame bound condition. History Because of the various mathematical components surrounding frames, frame theory has roots in harmonic and functional analysis, operator theory, linear algebra, and matrix theory. The Fourier transform has been used for over a century as a way of decomposing and expanding signals. However, the Fourier transform masks key information regarding the moment of emission and the duration of a signal. In 1946, Dennis Gabor was able to solve this using a technique that simultaneously reduced noise, provided resiliency, and created quantization while encapsulating important signal characteristics. This discovery marked the first concerted effort towards frame theory. The frame condition was first described by Richard Duffin and Albert Charles Schaeffer in a 1952 article on nonharmonic Fourier series as a way of computing the coefficients in a linear combination of the vectors of a linearly dependent spanning set (in their terminology, a "Hilbert space frame"). In the 1980s, Stéphane Mallat, Ingrid Daubechies, and Yves Meyer used frames to analyze wavelets. Today frames are associated with wavelets, signal and image processing, and data compression. Relation to bases A frame satisfies a generalization of Parseval's identity, namely the frame condition, while still maintaining norm equivalence between a signal and its sequence of coefficients. If the set is a frame of V, it spans V. Otherwise there would exist