希尔伯特空间中的近似对偶框架对及其对Gabor框架的应用
Approximately dual frame pairs in Hilbert spaces and applications to Gabor frames
摘要：
我们讨论了伪对偶框架和近似对偶框架的概念，并阐明了它们与经典框架的关系。近似对偶框架比经典对偶框架更容易构造，并且可以被定制以产生几乎完美的重建。对于通过扰动理论构造的近似对偶框架，我们给出了与完美重构的偏差的一个界。通过使用框架算子的Walnut表示来估计对偶条件下的不等式偏差，得到了富类Gabor框架的另一个界.作为结果的说明，我们构造了明确的近似的Gabor框架产生的高斯，这些近似的小波变换产生几乎完美的重建。令人惊讶的是，该方法也适用于某些远离紧的Gabor框架。
英文摘要：
We discuss the concepts of pseudo-dual frames and approximately dual frames, and illuminate their relationship to classical frames. Approximately dual frames are easier to construct than the classical dual frames, and might be tailored to yield almost perfect reconstruction. For approximately dual frames constructed via perturbation theory, we provide a bound on the deviation from perfect reconstruction. An alternative bound is derived for the rich class of Gabor frames, by using the Walnut representation of the frame operator to estimate the deviation from equality in the duality conditions. As illustration of the results, we construct explicit approximate duals of Gabor frames generated by the Gaussian; these approximate duals yield almost perfect reconstruction. Amazingly, the method applies also to certain Gabor frames that are far from being tight.
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