Exact recovery of non-uniform splines from the projection onto spaces of algebraic polynomials

TL;DR

This paper demonstrates that non-uniform splines can be exactly recovered from polynomial projections under certain conditions using Total Variation norm minimization, extending previous trigonometric polynomial results to algebraic polynomials and multivariate cases.

Contribution

It introduces a novel method for exact spline recovery from polynomial projections under a Chebyshev-type separation condition, generalizing existing theories to algebraic and multivariate splines.

Findings

Exact recovery of non-uniform splines under separation condition

Extension of dual polynomial method to algebraic polynomials

Results for multivariate spline recovery

Abstract

In this work we consider the problem of recovering non-uniform splines from their projection onto spaces of algebraic polynomials. We show that under a certain Chebyshev-type separation condition on its knots, a spline whose inner-products with a polynomial basis and boundary conditions are known, can be recovered using Total Variation norm minimization. The proof of the uniqueness of the solution uses the method of `dual' interpolating polynomials and is based on \cite{SR}, where the theory was developed for trigonometric polynomials. We also show results for the multivariate case.