# A Stability Barrier for Reconstructions from Fourier Samples

@article{Adcock2014ASB, title={A Stability Barrier for Reconstructions from Fourier Samples}, author={B. Adcock and A. Hansen and A. Shadrin}, journal={SIAM J. Numer. Anal.}, year={2014}, volume={52}, pages={125-139} }

We prove that any stable method for resolving the Gibbs phenomenon---that is, recovering high-order accuracy from the first $m$ Fourier coefficients of an analytic and nonperiodic function---can converge at best root-exponentially fast in $m$. Any method with faster convergence must also be unstable, and in particular, exponential convergence implies exponential ill-conditioning. This result is analogous to a recent theorem of Platte, Trefethen, and Kuijlaars concerning recovery from pointwise… CONTINUE READING

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