A Fundamental Identity for Parseval Frames

Frames are an essential tool for many emerging applications such as data transmission. Their main advantage is the fact that frames can be designed to be redundant while still providing reconstruction formulas. This makes them robust against noise and losses while allowing freedom in design (see, for example, [5, 10]). Due to their numerical stability, tight frames and Parseval frames are of increasing interest in applications (See Section 2.1 for definitions.). Particularly in image processing, tight frames have emerged as essential tool (compare [7]). In abstract frame theory, systems constituting tight frames and, in particular, Parseval frames have already been extensively explored [3, 5, 6, 9, 10, 11], yet many questions are still open. For many years engineers believed that, in applications such as speech recognition, a signal can be reconstructed without information about the phase. In [1] this longstanding conjecture was verified by constructing new classes of Parseval frames for which a signal vector can reconstructed without noisy phase or its estimation. While working on efficient algorithms for signal reconstruction, the authors of [1] discovered a surprising identity for Parseval frames (see [2] for a detailed discussion of the origins of the identity).