论文标题
伯格曼空间和分析小波的分形不确定性原理
A fractal uncertainty principle for Bergman spaces and analytic wavelets
论文作者
论文摘要
Motivated by results of Dyatlov on Fourier uncertainty principles for Cantor sets and by similar results of Knutsen for joint time-frequency representations (i.e., the short-time Fourier transform (STFT) with a Gaussian window, equivalent to Fock spaces), we suggest a general setting relating localization and uncertainty and prove, within this context, an uncertainty principle for Cantor sets in Bergman spaces on the unit disk, where the Cantor Set定义为Annuli的联合，在双曲线度量中等分。结果可以用分析性的Cauchy小波来编写。与Knutsen考虑的STFT一样，我们的结果包括针对指数中涉及分形尺寸log 2 / log 3的定位操作员的两侧结合。如在STFT情况和Dyatlov分形不确定性原理中一样，（双曲线）测量磁盘中的Cantor迭代率的（双曲线）倾向于无穷大，而定位算子的相应规范往往为零。
Motivated by results of Dyatlov on Fourier uncertainty principles for Cantor sets and by similar results of Knutsen for joint time-frequency representations (i.e., the short-time Fourier transform (STFT) with a Gaussian window, equivalent to Fock spaces), we suggest a general setting relating localization and uncertainty and prove, within this context, an uncertainty principle for Cantor sets in Bergman spaces on the unit disk, where the Cantor set is defined as a union of annuli that are equidistributed in the hyperbolic measure.The result can be written in terms of analytic Cauchy wavelets. As in the case of the STFT considered by Knutsen, our result consists of a two-sided bound for the norm of a localization operator involving the fractal dimension log 2 / log 3 in the exponent. As in the STFT case and in Dyatlov fractal uncertainty principle, the (hyperbolic) measure of the dilated iterates of the Cantor set in the disk tends to infinity, while the corresponding norm of the localization operator tends to zero.
