The Highly Oscillatory Behavior of Automorphic Distributions for SL(2)
The Highly Oscillatory Behavior of Automorphic Distributions for SL(2)
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DOI：
10.1007/s11005-004-0470-8
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发表时间：
2004-02
影响因子：
1.2
通讯作者：
S. Miller;W. Schmid
中科院分区：
文献类型：
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作者：
S. Miller;W. Schmid
SL(2)的自同构分布作为模形式的边值而产生，并且以更微妙的方式来自Maass形式。在权为1的模形式或Maass形式的情况下，自同构分布具有连续的一阶反导数。我们回顾了我们其中一人关于这些连续函数的Holder连续性的早期结果，并将它们与其他作者的结果联系起来；这涉及到S.Bernstein和Hardy和Littlewood关于傅立叶级数的经典定理的推广。然后，我们证明了反导数在所有无理性点，以及所有或在某些情况下，某些有理点是不可微的。我们包括了其中几个函数的图表，这些图表清楚地显示了高度的振荡。我们的研究部分是由“黎曼不可微函数”，也称为“魏尔斯特拉斯函数”的性质所推动的。
Automorphic distributions for SL(2) arise as boundary values of modular forms and, in a more subtle manner, from Maass forms. In the case of modular forms of weight one or of Maass forms, the automorphic distributions have continuous first antiderivatives. We recall earlier results of one of us on the Holder continuity of these continuous functions and relate them to results of other authors; this involves a generalization of classical theorems on Fourier series by S. Bernstein and Hardy and Littlewood. We then show that the antiderivatives are non-differentiable at all irrational points, as well as all, or in certain cases, some rational points. We include graphs of several of these functions, which clearly display a high degree of oscillation. Our investigations are motivated in part by properties of ‘Riemann’s nondifferentiable function’, also known as ‘Weierstrass’ function’.