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Description

Parallel to the development of wavelets and connected with it, there has arisen a simple idea in signal transmission to represent signals by their "Fourier" coefficients with respect to a redundant system of vectors (not orthogonal). The idea is that even if some (random) coefficients are lost on their way to the user, the signal can still be recovered from whatever received thanks to the redundancy in the coefficients. The hard question is -- how many random coefficients have to be received for successful recovery of the signal? This problem quickly reduced to a simple probabilistic claim, which is hard to prove though. Unexpectedly, a sharp approach is possible through the non-commutative operator theory. This method was suggested by Pisier developed by Rudelson and later by the PI and Rudelson.