分段D-有限函数的矩反演问题
Moment inversion problem for piecewise D-finite functions
摘要：
本文研究了在未知位置具有跳跃间断的一元函数的精确重构问题。这些函数被假定为满足一个先验未知的线性齐次微分方程的多项式系数的每个连续区间。因此，它们可以由有限的信息量来指定。该重构问题在信号处理和其它应用中具有实际意义。这是一个有点“民间传说”的序列的时刻，这样的“分段D-有限”功能满足一个线性递归关系的有界秩序和程度。我们明确地推导出这个递归关系。原来，消除函数的每一部分的微分算子的系数，以及不连续的位置，以精确控制的方式出现在这个递归中。这导致制定一个通用的算法重建分段D-有限的功能，从它的时刻。我们研究了一般情况下所得到的线性系的可解性条件，并分析了几个特殊的例子。我们提供的数值模拟的几种类型的信号，测试所提出的算法对噪声的敏感性的结果。
英文摘要：
We consider the problem of exact reconstruction of univariate functions with jump discontinuities at unknown positions from their moments. These functions are assumed to satisfy an a priori unknown linear homogeneous differential equation with polynomial coefficients on each continuity interval. Therefore, they may be specified by a finite amount of information. This reconstruction problem has practical importance in Signal Processing and other applications. It is somewhat of a ``folklore'' that the sequence of the moments of such ``piecewise D-finite''functions satisfies a linear recurrence relation of bounded order and degree. We derive this recurrence relation explicitly. It turns out that the coefficients of the differential operator which annihilates every piece of the function, as well as the locations of the discontinuities, appear in this recurrence in a precisely controlled manner. This leads to the formulation of a generic algorithm for reconstructing a piecewise D-finite function from its moments. We investigate the conditions for solvability of the resulting linear systems in the general case, as well as analyze a few particular examples. We provide results of numerical simulations for several types of signals, which test the sensitivity of the proposed algorithm to noise.
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