Some iterative algorithms to compute canonical windows for Gabor frames

We analyze some iterative algorithms for the computation of the canonical tight window g^t and the canonical dual window g^d associated with a Gabor frame (g; a; b). As to the computation of g^t, we consider algorithms that do require inversion of intermediate frame operators as well as algorithms that do not require inversions. As to the computation of g^d, we naturally consider algorithms where no frame operator inversions are required. These algorithms have safe but conservative versions, with guaranteed convergence of prescribed order but with suboptimal convergence constants, and smart but risky versions, with near-optimal convergence of prescribed order which is, however, guaranteed only if the frame bound ratio A=B of (g; a; b) exceeds an analytically given lower bound. Thus we propose for g^t an algorithm, using inversions, with quadratic convergence, and two algorithms, using no inversions, with quadratic and cubic convergence, respectively, and we identify for these algorithms the safe and the smart versions. For g^d we propose two algorithms, without inversions, with quadratic and cubic convergence, respectively, and also for these algorithms we identify the safe and the smart versions. All these algorithms can be formulated by using a general mechanism for proposing the recursion step in an approximation scheme for g^t and g^d with a prescribed error decay. The tools used to analyze the algorithms are the calculus of frame operators, the spectral mapping theorem, and Kantorovich’s inequality.