Stefan Kunis

The truncated multidimensional moment problem is studied as an interpolation problem on the special sets, which will be defined in this paper. The step-by-step algorithm is obtained and description of all solutions in symmetric form is found.

We study signals that are sparse either on the vertices of a graph or in the graph spectral domain. Recent results on the algebraic properties of random integer matrices as well as on the boundedness of eigenvectors of random matrices imply two types of support size uncertainty principles for graph signals. Indeed, the algebraic properties imply uniqueness results if a sparse signal is sampled at any set of minimal size in the other domain. The boundedness properties of eigenvectors imply stable reconstruction by basis pursuit if a sparse signal is sampled at a slightly larger randomly selected set in the other domain.

This paper considers superresolution (SR) as the mapping of Fourier coefficients of a discrete measure on [Formula: see text] to its support and weights. We focus on weak SR assuming a condition on the separation of the involved measures similar to the Rayleigh criterion and prove that the reconstruction map satisfies a local Lipschitz property giving explicit estimates for the Lipschitz constant depending on the dimension [Formula: see text] and the sampling effort while improving a bound on the assumed separation from [ 6 ]. With the Wasserstein distance as the metric on the space of measures we even conclude that weak SR is globally Lipschitz continuous and hence well-conditioned.

Classical fluorescence microscopy is a powerful technique to image biological specimen under close-to-native conditions, but light diffraction limits its optical resolution to 200–300 nm-two orders of magnitude worse than the size of biomolecules. Assuming single fluorescent emitters, the final image of the optical system can be described by a convolution with the point spread function (PSF) smearing out details below the size of the PSF. In mathematical terms, fluorescence microscopy produces bandlimited space-continuous images that can be recovered from their spatial samples under the conditions of the classical Shannon-Nyquist theorem. During the past two decades, several single molecule localization techniques have been established and these allow for the determination of molecular positions with sub-pixel accuracy. Without noise, single emitter positions can be recovered precisely – no matter how close they are. We review recent work on the computational resolution limit with a sharp phase transition between two scenarios: 1) where emitters are well-separated with respect to the bandlimit and can be recovered up to the noise level and 2) closely distributed emitters which results in a strong noise amplification in the worst case. We close by discussing additional pitfalls using single molecule localization techniques based on structured illumination.

Complex signed measures of finite total variation are a powerful signal model in many applications. Restricting to the $d$-dimensional torus, finitely supported measures allow for exact recovery if the trigonometric moments up to some order are known. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the Wasserstein-1 distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known. A second class of sum of squares polynomials is shown to interpolate the characteristic function on the support of the measure and to converge to zero outside.

Estimates on the condition number of Vandermonde matrices have implications on several algorithms ranging from polynomial interpolation to sparse super resolution in fluorescence microscopy. Classically, the situation is studied for monomials on real intervals, the complex unit disk, and the complex unit circle. Except for roots of unity and well separated nodes on the unit circle, the condition number grows strongly with increasing polynomial degree. Here, we show that the condition number of the Vandermonde matrix for a particular instance of critically separated nodes on the complex unit circle grows logarithmically with the polynomial degree. The proof is based on a variant of Hilbert's inequality with remainder term.

While structured illumination microscopy (SIM) is a widely used super‐resolution microscopy technique for densely labelled samples, we apply a modified version of the matrix pencil method to single molecule imaging. We prove that under reasonable conditions we can recover up to two times more molecule positions compared to conventional microscopy exactly from noiseless data and include numerical examples illustrating benefits as well as limitations of the algorithm.