代数信号采样、吉布斯现象与普朗尼型系统
Algebraic signal sampling, Gibbs phenomenon and Prony-type systems
AI总结：
本文提出了一种解决普朗尼型系统的新方法，通过在算术级数上采样信号，保持方程数量小且固定，从而提高重建精度，并解决了Eckhoff猜想，消除吉布斯现象。
摘要：
普朗尼型系统出现在各种信号重建问题中，如有限创新率、超分辨率和傅里叶反演分段光滑函数。本文提出了一种新的解决方法，要求在算术级数上采样信号。通过保持方程数量小且固定，证明这种“降采样”可以带来实际的重建精度提升。作为应用，本文提供了解决所谓的Eckhoff猜想的方法，该猜想要求从傅里叶系数中以可能的最大渐近精度重建分段光滑函数的跃迁位置和幅度，从而消除吉布斯现象。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
