  We describe a family of iterative algorithms that involve the repeated
execution of discrete and inverse discrete Fourier transforms. One interesting
member of this family is motivated by the discrete Fourier transform
uncertainty principle and involves the application of a sparsification
operation to both the real domain and frequency domain data with convergence
obtained when real domain sparsity hits a stable pattern. This sparsification
variant has practical utility for signal denoising, in particular the recovery
of a periodic spike signal in the presence of Gaussian noise. General
convergence properties and denoising performance relative to existing methods
are demonstrated using simulation studies. An R package implementing this
technique and related resources can be found at
https://hrfrost.host.dartmouth.edu/IterativeFT.
