Algebraic Fourier reconstruction of piecewise smooth functions

Abstract

Abstract. Accurate reconstruction of piecewise-smooth functions from a finite number of Fourier coefficients is an important problem in various applications. This probelm exhibits an inherent inaccuracy, in particular the Gibbs phenomenon, and it is being intensively investigated during the last decades. Several nonlinear reconstruction methods have been proposed in the literature, and it is by now well-established that the "classical" convergence order can be completely restored up to the discontinuities. Still, the maximal accuracy of determining the positions of these discontinuities remains an open question.In this paper we prove that the locations of the jumps (and subsequently the pointwise values of the function) can be reconstructed with at least "half the classical accuracy". In particular, we develop a constructive approximation procedure which, given the first Fourier coefficients of a piecewise-¾ ·½ function, recovers the locations of the jumps with accuracy ´ ·¾µ , and the values of the function between the jumps with accuracy ´ ·½µ (similar estimates are obtained for the associated jump magnitudes). A key ingredient of the algorithm is to start with the case of a single discontinuity, where a modified version of one of the existing algebraic methods (due to K.Eckhoff) may be applied. It turns out that the additional orders of smoothness produce highly correlated error terms in the Fourier coefficients, which eventually cancel out in the corresponding algebraic equations. To handle more than one jump, we apply a localization procedure via a convolution in the Fourier domain, which eventually preserves the accuracy estimates obtained for the single jump. We provide some numerical results which support the theoretical predictions.

Towards a Mathematical Theory of Super‐resolution

“…Whereas we have tried to optimize the constant in one dimension, 2 we have not really attempted to do so here in order to keep the proof reasonably short and simple. Hence this theorem is subject to improvement.…”

“…As a result, it is highly unstable in the presence of noise as discussed in [38] and in the presence of approximate sparsity. Algebraic techniques have also been applied to the location of singularities in the reconstruction of piecewise polynomial functions from a finite number of Fourier coefficients (see [1,2,19] and references therein). The theoretical analysis of these methods proves their accuracy up to a certain limit related to the number of measurements.…”

Complex Analysis and Dynamical Systems V

“…This includes linear combinations of shifts of known functions, signals with "finite rate of innovation", piecewise D-finite functions which we use below, piecewise-smooth functions, and many other cases (see [5,6,9,14,35] and references therein).…”

Sampling Piecewise Sinusoidal Signals With Finite Rate of Innovation Methods

“…Then the equivalent kernel ϕ ′ (t) is of compact support W ′ = W + 2DN . Furthermore, according to (13), x ′ (t) is made only of differentiated Diracs of maximum order 2DN − 1 in the discontinuities. That is, we are left with a signal of the type x ′ (t) = D+1 d=1…”

“…The reconstruction process for these schemes is based on the annihilating filter method, a tool widely used in spectral estimation [9], error correction coding [10], interpolation [11] and for solving inverse problems [12], [13], [14], [15]. These results provide an answer for precise time localization (i.e.…”