Convolution powers of complex functions on Z d
Convolution powers of complex functions on Z d
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Z d 上复函数的卷积幂
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
M. V. Fedoryuk
中科院分区：
文献类型：
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作者：
Evan Randles;L. Saloff‐Coste;V. Thomée;M. V. Fedoryuk
d维正方形格点上概率分布φ的卷积幂是随机游动理论的核心。例如，第n次卷积幂φ是相关随机游走的第n步的分布，并由经典局部极限定理描述。在P. Diaconis和作者以前的工作之后，我们探索了φ取复数值的更一般的设置。本研究的出发点是Erastus L.德福雷斯特在数据平滑方面的研究，已经应用于偏微分方程数值差分格式的稳定性理论。对于Z上的复值函数φ，我们提出并解决了关于卷积幂φ的四个基本问题，它们涉及超范数估计、广义局部极限定理、逐点估计和稳定性。这一工作推广了I. J. Schoenberg，T. N. E. Greville，P.Diaconis和第二作者的结果，以及在稳定性理论的背景下，V.V.费多留克
The study of convolution powers of a finitely supported probability distribution φ on the d-dimensional square lattice is central to random walk theory. For instance, the nth convolution power φ is the distribution of the nth step of the associated random walk and is described by the classical local limit theorem. Following previous work of P. Diaconis and the authors, we explore the more general setting in which φ takes on complex values. This exploration, originally motivated by the problem of Erastus L. De Forest in data smoothing, has found applications to the theory of stability of numerical difference schemes in partial differential equations. For a complex valued function φ on Z, we ask and address four basic and fundamental questions about the convolution powers φ which concern sup-norm estimates, generalized local limit theorems, pointwise estimates, and stability. This work extends one-dimensional results of I. J. Schoenberg, T. N. E. Greville, P. Diaconis and the second author and, in the context of stability theory, results by V. Thomée and M. V. Fedoryuk.