Super-Resolution of Positive Sources on an Arbitrarily Fine Grid

In super-resolution it is necessary to locate with high precision point sources from noisy observations of the spectrum of the signal at low frequencies capped by flo\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{\mathrm {lo}}$$\end{document}. In the case when the point sources are positive and are located on a grid, it has been recently established that the super-resolution problem can be solved via linear programming in a stable manner and that the method is nearly optimal in the minimax sense. The quality of the reconstruction critically depends on the Rayleigh regularity of the support of the signal; that is, on the maximum number of sources that can occur within an interval of side length about 1/flo\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/f_{\mathrm {lo}}$$\end{document}. This work extends the earlier result and shows that the conclusion continues to hold when the locations of the point sources are arbitrary, i.e., the grid is arbitrarily fine. The proof relies on new interpolation constructions in Fourier analysis.