Short background to question: Maybe one of the most famous mathematical transforms is the Fourier Transform, which has countless applications across all possible sciences and engineering branches. One very important discovery was the Fast Fourier Transform (FFT) which made it possible to calculate Discrete Fourier Transforms (DFT) faster than ever before.
The Fourier transform is a unitary transform. This means energy is preserved. The basis functions span a basis for the space which is investigated. This allows for one unique representation. However it is not always unproblematic. For example if we want to represent a sine wave which has a frequency which is a non integer fraction of the fundamental frequency of the FFT, we will not be able to represent it perfectly. This is well known among for example signal and audio engineers. But what we can do is to add this sine wave of odd frequency as a vector. What we then get is a frame or an overcomplete representation of all possible functions we can transform.
Now to the question, does there exist any systematic way to construct Fourier Frames in a way which can somehow utilize the great speed gains of the FFT? In other words, Fast Fourier Frames (FFF), are they possible?