How are generalized frames related to biorthogonal bases? It seems like frames are a possible solution if neither orthonormal nor biorthogonal bases are available. I thought the generalized frames theory was what enabled the use of biorthogonal bases and the redundancy it gives.
Edit: It seems like biorthogonal bases might be a special case of frames and frames are even more general than I thought, frames are not only used with wavelets.
I found the answer in Jelena Kovacevic excellent article Life Beyond Bases: The Advent of Frames (Part 1), page 90-91, IEEE Signal Processing Magazine, July 2007. When you have more bases vectors than dimensions the resulting (redundant) set (of linearly dependent vectors) is called a frame instead of a basis. The frame has a dual frame and biorthogonal bases is a special (case) frame where the frame and the dual frame are interchangeable. x=FDx=DFx where F is the frame, D is the dual frame and * is the Hermitian transposition.
"Bi essentially just means two. In bilinear it means linear in two arguments. In biorthogonal it means two families of vectors are orthogonal in respect to one another (but neither must be orthogonal in respect to itself)." The inner product of the n:th position from each vector family is zero. Does the "bi" in bilinear and biorthogonal mean different things?