Algebraic signal sampling, Gibbs phenomenon and Prony-type systems

TL;DR

This paper introduces a new sampling method for Prony-type systems that enhances signal reconstruction accuracy, successfully addressing the Gibbs phenomenon and solving Eckhoff's conjecture on reconstructing piecewise-smooth functions from Fourier data.

Contribution

It proposes a decimation-based sampling approach for Prony systems, improving reconstruction precision and providing a solution to Eckhoff's conjecture.

Findings

Decimation improves reconstruction accuracy in Prony systems.

The method effectively eliminates the Gibbs phenomenon.

It achieves maximal asymptotic accuracy in reconstructing piecewise-smooth functions.

Abstract

Systems of Prony type appear in various signal reconstruction problems such as finite rate of innovation, superresolution and Fourier inversion of piecewise smooth functions. We propose a novel approach for solving Prony-type systems, which requires sampling the signal at arithmetic progressions. By keeping the number of equations small and fixed, we demonstrate that such "decimation" can lead to practical improvements in the reconstruction accuracy. As an application, we provide a solution to the so-called Eckhoff's conjecture, which asked for reconstructing jump positions and magnitudes of a piecewise-smooth function from its Fourier coefficients with maximal possible asymptotic accuracy -- thus eliminating the Gibbs phenomenon.