and has positive upper and lower bounds. The inverse of exists and it, too, is self-adjoint, positive definite, and has positive upper and lower bounds. The dual frame is defined by mapping each element of the frame with : To see that this makes sense, let be an element of and let Thus which proves that Alternatively, we can let By inserting the above definition of and applying the properties of and its inverse, which shows that The numbers are called frame coefficients. This derivation of a dual frame is a summary of Section 3 in the article by Duffin and Schaeffer. They use the term conjugate frame for what here is called a dual frame. The dual frame is called the canonical dual of because it acts similarly as a dual basis to a basis. When the frame is overcomplete, a vector can be written as a linear combination of in more than one way. That is, there are different choices of coefficients such that This allows us some freedom for the choice of coefficients other than It is necessary that the frame is overcomplete for other such coefficients to exist. If so, then there exist frames for which for all We call a dual frame of Canonical duality is a reciprocity relation, i.e. if the frame is the canonical dual frame of then is the canonical dual frame of See also k-frame Biorthogonal wavelet Orthogonal wavelet Restricted isometry property Schauder basis Harmonic analysis Fourier analysis Functional analysis Notes References Linear algebra Differential geometry Signal processing