斜对偶性在框架和概率框架中的土地
A Land of Oblique Duality for Frames and Probabilistic Frames
AI总结：
本文提出斜对偶概率框架的概念，通过分析其势能函数，证明了在特定条件下框架的紧致性和斜对偶的标准化特性。
摘要：
用于采样和重建信号的函数或分布经常出现在不同的领域，例如经典采样中的狄拉克δ函数和带限的脉冲函数。斜对偶框架推广了这一现象。在本文中，我们提供新的工具来研究斜对偶框架，并引入了斜对偶概率框架的变体。我们首先提出了斜对偶框架势能，并证明当斜对偶与标准斜对偶相同时，该势能被最小化。然后我们定义了斜对偶概率框架和斜近似双概率框架。特别地，我们证明，对于给定的斜对偶概率框架，其对应的斜对偶概率框架势能被最小化当且仅当框架是紧致的且斜对偶是标准的。此外，当限制在推前类型的斜对偶概率框架上时，可以去除紧致性假设。最后，我们研究了斜对偶概率框架的扰动，并证明如果概率测度在2-瓦瑟斯坦拓扑下足够接近于斜对偶概率框架对，则它会形成一个斜近似双概率框架。
英文摘要：
Functions or distributions used to sample and to reconstruct signals often occur in different domains, like the Dirac delta and a band-limited bump function in classical sampling. Oblique dual frames generalize this phenomenon. In this paper, we provide new tools to study oblique dual frames and introduce a probabilistic variant of oblique dual frames. We first present the oblique dual frame potential and show that it is minimized precisely when the oblique dual coincides with the canonical oblique dual. We then define oblique dual probabilistic frames and oblique approximately dual probabilistic frames. In particular, we prove that for a given oblique dual probabilistic frame, the associated oblique dual probabilistic frame potential is minimized if and only if the frame is tight and the oblique dual is canonical. Moreover, the tightness assumption can be removed when the minimization is restricted to oblique dual probabilistic frames of pushforward type. Finally, we investigate perturbations of oblique dual probabilistic frames and show that if a probability measure is sufficiently close to an oblique dual probabilistic frame pair in the $2$-Wasserstein topology, then it forms an oblique approximately dual probabilistic frame.
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