In this paper we formulate Donoho and Logan's large sieve principle for the wavelet transform on the Hardy space, adapting the concept of maximum Nyquist density to the hyperbolic geometry of the underlying space. The results provide deterministic guarantees for L1-minimization methods and hold for a class of mother wavelets that constitutes an orthonormal basis of the Hardy space and can be associated with higher hyperbolic Landau levels. Explicit calculations of the basis functions reveal a connection with the Zernike polynomials. We prove a novel local reproducing formula for the spaces in consideration and use it to derive concentration estimates of the large sieve type for the corresponding wavelet transforms. We conclude with a discussion of optimality of localization and Lieb inequalities in the analytic case by building on recent results of Kulikov, Ramos and Tilli based on the groundbreaking methods of Nicola and Tilli. This leads to a sharp uncertainty principle and a local Lieb inequality for the wavelet transform.

中文翻译：

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用于小波变换的 Donoho-Logan 大筛原则


在本文中，我们制定了 Donoho 和 Logan 的大筛原理，用于 Hardy 空间上的小波变换，将最大奈奎斯特密度的概念应用于底层空间的双曲几何。结果为 L1 最小化方法提供了确定性保证，并且适用于一类母小波，该小波构成了 Hardy 空间的正交基，并且可以与更高的双曲 Landau 水平相关联。基函数的显式计算揭示了与 Zernike 多项式的联系。我们为所考虑的空间证明了一种新的局部再现公式，并使用它来推导出相应小波变换的大筛类型的浓度估计。最后，我们以 Kulikov、Ramos 和 Tilli 基于 Nicola 和 Tilli 的开创性方法的最新结果为基础，讨论了分析案例中的局部最优性和 Lieb 不等式。这导致了尖锐的不确定性原理和小波变换的局部 Lieb 不等式。