Quadratic Weyl sums, automorphic functions and invariance principles
Quadratic Weyl sums, automorphic functions and invariance principles
复制标题
二次韦尔和、自守函数和不变性原理
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
J. Marklof
中科院分区：
文献类型：
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作者：
F. Cellarosi;J. Marklof
哈代和Littlewood的二次Weyl和（θ和）的近似函数方程，通过迭代应用，提供了一个强有力的工具，这种总和的渐近分析。另一方面，经典的Jacobi theta函数满足一个精确的函数方程，并扩展到Jacobi群上的自守函数。在本研究中，我们构造了一个相关的，几乎处处不可微的自守函数，它近似二次外尔和，直到一阶误差，在求和范围内一致。这不仅意味着近似的函数方程，但也允许我们取代哈代和Littlewood的重整化方法的动力学的一个特定的均匀流。这种构造的最大优点是近似是全局的，也就是说，不需要跟踪在迭代过程中积累的误差项。我们的主要应用是一个新的功能极限定理，或不变原理，θ总和。这里有趣的观察是，极限过程的路径与布朗运动共享许多关键特征（尺度不变性，时间反演下的不变性和不可微性），尽管时间增量不是独立的，并且每个固定时间的值分布与正态分布明显不同。
Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non‐differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but also allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, that is, there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation here is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion and non‐differentiability), although time increments are not independent and the value distribution at each fixed time is distinctly different from a normal distribution.
影响因子：
2.3
作者：
Melbourne, Ian;Nicol, Matthew
通讯作者：
Nicol, Matthew