Non-uniform spline recovery from small degree polynomial approximation
Provides theoretical guarantees for grid-free recovery of spline knots from noisy polynomial measurements, relevant to signal processing and approximation theory.
The paper studies sparse spikes deconvolution in algebraic polynomial spaces, proving that knots of non-uniform splines can be recovered via semidefinite programming under a Chebyshev-type separation condition, with quantitative bounds on support and amplitude errors.
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.