Introduction to Finite Frame Theory

The Fourier transform has been a major tool in analysis for over 100 years. However, it solely provides frequency information, and hides (in its phases) information concerning the moment of emission and duration of a signal. D. Gabor resolved this problem in 1946 [93] by introducing a fundamental new approach to signal decomposition. Gabor’s approach quickly became the paradigm for this area, because it provided resilience to additive noise, resilience to quantization, resilience to transmission losses as well as an ability to capture important signal characteristics. Unbeknownst to Gabor, he had discovered the fundamental properties of a frame without any of the formalism. In 1952, Duffin and Schaeffer [80] were working on some deep problems in non-harmonic Fourier series for which they required a formal structure for working with highly over-complete families of exponential functions in L2[0,1]. For this, they introduced the notion of a Hilbert space frame, for which Gabor’s approach is now a special case, falling into the area of timefrequency analysis [98]. Much later – in the late 1980’s – the fundamental concept of frames was revived by Daubechies, Grossman and Mayer [77] (see also [76]), who showed its importance for data processing. Traditionally, frames were used in signal and image processing, non-harmonic Fourier series, data compression, and sampling theory. But today, frame theory has ever increasing applications to problems in both pure and applied mathematics,