The problem of computational super-resolution asks to recover an object from its noisy and limited spectrum. In this talk, we consider two inverse problems of this flavor, mainly from the point of view of stability estimates. In the first problem, we assume that the object's spectrum is a finite sum of exponentials modulated by polynomials (extending the well-researched case where the polynomials are constants). We derive upper bounds on the problem condition number and show that the attainable resolution exhibits Hölder-type continuity with respect to the noise level [1,3]. As an application we consider the approximation of a piecewise-smooth function from its Fourier coefficients. We can show that the asymptotic accuracy of our approach is only dictated by the smoothness of the function between the jumps, even if the jump locations are not known [2].The second problem is concerned with on-going work on the weighted extrapolation problem on the real line for functions of finite exponential type where we abandon the sparsity assumption. It turns out that the extrapolation range scales logarithmically with the noise level, while the pointwise extrapolation error exhibits again a Hölder-type continuity.
References:
[1] A. Akinshin, D. Batenkov, and Y. Yomdin, “Accuracy of spike-train Fourier reconstruction for colliding nodes,” in 2015 International Conference on Sampling Theory and Applications (SampTA), 2015, pp. 617–621.
[2] D. Batenkov, “Complete algebraic reconstruction of piecewise-smooth functions from Fourier data,” Math. Comp., vol. 84, no. 295, pp. 2329–2350, 2015
[3] D. Batenkov, “Stability and super-resolution of generalized spike recovery,” Applied and Computational Harmonic Analysis, http://dx.doi.org/10.1016/j.acha.2016.09.004.