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We investigate the multi-dimensional super resolution problem on closed semi-algebraic domains for various sampling schemes such as Fourier or moments. We present a new semidefinite programming (SDP) formulation of the \ell 1 -minimization in the space of Radon measures in the multi-dimensional frame on semi-algebraic sets. While standard approaches have focused on SDP relaxations of the dual program (a popular approach is based on Gram matrix representations), this paper introduces an exact formulation of the primal \ell 1 -minimization exact recovery problem of super resolution that unleashes standard techniques (such as moment-sum-of-squares hierarchies) to overcome intrinsic limitations of previous works in the literature. Notably, we show that one can exactly solve the super resolution problem in dimension greater than 2 and for a large family of domains described by semi-algebraic sets. [ABSTRACT FROM PUBLISHER]

In this paper, by using the spectral analysis of the relevant linear operator and Gelfand’s formula, we obtain some properties of the first eigenvalue of a fractional differential equation. Based on these properties, the fixed point index of the nonlinear operator is calculated explicitly and some sufficient conditions for the existence of positive solutions are established. [ABSTRACT FROM AUTHOR]

This article investigates the support detection problem using the LASSO estimator in the space of measures. More precisely, we study the recovery of a discrete measure (spike train) from few noisy observations (Fourier samples, moments, etc.) using an ℓ 1 -regularization procedure. In particular, we provide an explicit quantitative localization of the spikes. [ABSTRACT FROM AUTHOR]

Abstract: We show that there exist bivariate proper quasi-copulas that do not induce a doubly stochastic signed measure on . We construct these quasi-copulas from the so-called proper quasi-transformation square matrices. [Copyright &y& Elsevier]