论文标题
$ \ mathbb {z}^d $上的复杂函数的本地限制定理
Local Limit Theorems for Complex Functions on $\mathbb{Z}^d$
论文作者
论文摘要
本地（中央）限制定理精确地描述了$ d $二维整数晶格上概率分布的迭代卷积幂的行为，$ \ mathbb {z}^d $。在某些对分布的温和假设下，定理说，卷积能力被单个缩放的高斯密度良好，我们称为吸引子。当允许这样的分布进行复杂的价值时，它们的卷积能力将表现出在概率环境中看不见的新行为和不同的行为。遵循I. J. Schoenberg，T。N。E. Greville，P。Diaconis和L. Saloff-Coste的作品，作者和L. Saloff-Coste提供了有关$ \ \ MATHBB {Z} $的有限支持的复杂值函数的局部极限定理的完整描述。对于$ \ mathbb {z}^d $上的复杂值函数的卷积功能，鲜为人知。在作者和L. Saloff-Coste的先前工作中，为复杂值的功能建立了局部限制定理，其傅立叶变换的绝对值在所谓的正均匀类型的所谓点上被最大化，在这种情况下，在这种情况下，所得的吸引子是通用的热量内核，与一类高阶差分差异的运营商相应。通过考虑可以在虚构类型的点上以绝对值最大化傅立叶变换的可能性，本文扩展了作者的先前工作和L. saloff-coste，以扩大可以获得局部限制定理的复杂值函数类别。这些局部限制定理包含某些振荡积分给出的吸引子，其收敛是使用H. bui和作者以及van der corput引理的广义极性集成公式建立的。该文章还扩展了有关H. Bui和作者的SUP-NORM类型估计的最新结果。
The local (central) limit theorem precisely describes the behavior of iterated convolution powers of a probability distribution on the $d$-dimensional integer lattice, $\mathbb{Z}^d$. Under certain mild assumptions on the distribution, the theorem says that the convolution powers are well-approximated by a single scaled Gaussian density which we call an attractor. When such distributions are allowed to take on complex values, their convolution powers exhibit new and disparate behaviors not seen in the probabilistic setting. Following works of I. J. Schoenberg, T. N. E. Greville, P. Diaconis, and L. Saloff-Coste, the author and L. Saloff-Coste provided a complete description of local limit theorems for the class of finitely supported complex-valued functions on $\mathbb{Z}$. For convolution powers of complex-valued functions on $\mathbb{Z}^d$, much less is known. In a previous work by the author and L. Saloff-Coste, local limit theorems were established for complex-valued functions whose Fourier transform is maximized in absolute value at so-called points of positive homogeneous type and, in that case, the resultant attractors are generalized heat kernels corresponding to a class of higher order partial differential operators. By considering the possibility that the Fourier transform can be maximized in absolute value at points of imaginary homogeneous type, this article extends previous work of the author and L. Saloff-Coste to broaden the class of complex-valued functions for which it is possible to obtain local limit theorems. These local limit theorems contain attractors given by certain oscillatory integrals and their convergence is established using a generalized polar-coordinate integration formula, due to H. Bui and the author, and the Van der Corput lemma. The article also extends recent results on sup-norm type estimates of H. Bui and the author.
