  We investigate the sparse spikes deconvolution problem onto spaces of
algebraic polynomials. Our framework encompasses the measure reconstruction
problem from a combination of noiseless and noisy moment measurements. We study
a TV-norm regularization procedure to localize the support and estimate the
weights of a target discrete measure in this frame. Furthermore, we derive
quantitative bounds on the support recovery and the amplitudes errors under a
Chebyshev-type minimal separation condition on its support. Incidentally, we
study the localization of the knots of non-uniform splines when a Gaussian
perturbation of their inner-products with a known polynomial basis is observed
(i.e. a small degree polynomial approximation is known) and the boundary
conditions are known. We prove that the knots can be recovered in a grid-free
manner using semidefinite programming.
