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Coupling the periodic time-invariance of the wavelet transform, with a view to thresholding as a projection, yields a simple, recursive, wavelet-based technique for denoising signals. Estimating a signal from a noise-corrupted observation is a fundamental problem of signal processing which has been addressed via many techniques. Previously, R.R. Coifman and D.L. Donoho (see Wavelets and Statistics, Lecture Notes in Statistics, vol.103, p.125-50, 1995) introduced cycle spinning, a technique of estimating the true signal as the linear average of individual estimates derived from wavelet-thresholded translated versions of the noisy signal. We demonstrate that such an average can be improved upon dramatically. The proposed algorithm recursively "cycle spins" by repeatedly translating and denoising the input via basic wavelet denoising and then translating back; at each iteration, the output of the previous iteration is used as input. Exploiting the convergence properties of projections, our algorithm can be regarded as a sequence of denoising projections that converge to the projection of the original noisy signal to a small subspace containing the true signal. It is proven that the algorithm is guaranteed to converge globally, and simulations on piecewise polynomial signals show marked improvement over both basic wavelet thresholding and standard cycle spinning.