Geometry of Hilbert Space Frames

In applied linear algebra, the term frame is used to refer to a redundant or linearly dependent coordinate system. The concept was introduced in the study of Fourier series and is pertinent in signal processing, where the reconstruction property for finite frames allows for redundant transmission of data to guard against losses due to noise. We give a brief introduction to the theory of finite frames in Section 1, including the major results that allow for the easy construction and description of frames. The subsequent sections relate to the theoretical importance of frames. As a natural extension of the definition of basis, we are lead naturally to ask the same topological questions for the space of frames as we do for the space of bases, GLn(R). Two particular questions are explored in Sections 2 and 3. The reconstruction property for finite frames leads to a natural generalization in the realm of measure theory. This is the scope of Section 4, culminating in the approximation theorem for “frame measures”