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We extend recent results by Pisier on$K$-subcouples, i.e. subcouples of an interpolation couple that preserve the$K$-functional (up to constants) and corresponding notions for quotient couples. Examples include interpolation (in the pointwise sense) and a reinterpretation of the Adamyan-Arov-Krein theorem for Hankel operators.

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We study the super-resolution (SR) problem of recovering point sources consisting of a collection of isolated and suitably separated spikes from only the low frequency measurements. If the peak separation is above a factor in $(1, 2)$ of the Rayleigh length (physical resolution limit), L1 minimization is guaranteed to recover such sparse signals. However, below such critical length scale, especially the Rayleigh length, the $L_1$ certificate no longer exists. We show several local properties (local minimum, directional stationarity, and sparsity) of the limit points of minimizing two $L_1$ based nonconvex penalties, the difference of $L_1$ and $L_2$ norms $(L_{1-2})$ and capped $L_1 (CL_1)$, subject to the measurement constraints. In one and two dimensional numerical SR examples,the local optimal solutions from dierence of convex function algorithms outperform the global L1 solutions near or below Rayleigh length scales either in the accuracy of ground truth recovery or in nding a sparse solution satisfying the constraints more accurately.

Wave propagation problems arise in a wide range of applications. The energy conserving property is one of the guiding principles for numerical algorithms, in order to minimize the phase or shape errors after long time integration. In this paper, we develop and analyze a local discontinuous Galerkin (LDG) method for solving the wave equation. We prove optimal error estimates, superconvergence toward a particular projection of the exact solution, and the energy conserving property for the semi-discrete formulation. The analysis is extended to the fully discrete LDG scheme, with the centered second-order time discretization (the leap-frog scheme). Our numerical experiments demonstrate optimal rates of convergence and superconvergence. We also show that the shape of the solution, after long time integration, is well preserved due to the energy conserving property.