So far there has not been paid attention in the literature to frames that are balanced, i.e. those frames which sum is zero. In this paper we consider balanced frames, and in particular balanced unit norm tight frames in finite dimensional Hilbert spaces. Unit norm tight frames play a central role in frame theory and its applications. Here we discover the various advantages of balanced unit norm tight frames in signal processing and show that they turn out to perform better than the non balanced ones. They give an exact reconstruction in the presence of systematic errors in the transmitted coefficients, and are optimal when these coefficients are corrupted with noises that can have non-zero mean. Moreover, using balanced frames we can easily know that the transmitted coefficients were perturbed, and we also have an indication of the source of the error. We analyze several properties of these types of frames. In particular, we define an equivalence relation in the set of the dual frames of a balanced frame, and use it to show that we can obtain easily all the duals from the balanced ones. We study the problems of finding the nearest balanced frame in the $\ell^{1}$ and $\ell^{2}$ norms to a given frame, characterizing completely its existence and giving its expression. Finally, we present many examples and methods for constructing balanced unit norm tight frames.

中文翻译：

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平衡帧：具有良好特性的信号处理中的有用工具

迄今为止，文献中还没有关注平衡的帧，即总和为零的那些帧。在本文中，我们考虑平衡框架，特别是有限维希尔伯特空间中的平衡单位范数紧框架。单位范数紧框架在框架理论及其应用中起着核心作用。在这里，我们发现了平衡单元范数紧帧在信号处理中的各种优势，并表明它们比非平衡的表现更好。它们在传输系数中存在系统误差的情况下给出精确的重建，并且当这些系数被可能具有非零均值的噪声破坏时是最佳的。此外，使用平衡帧，我们可以很容易地知道传输系数受到了扰动，我们也有错误来源的指示。我们分析了这些类型框架的几个属性。特别地，我们在平衡坐标系的对偶坐标系的集合中定义了一个等价关系，并用它来表明我们可以很容易地从平衡坐标系中获得所有对偶。我们研究了在 $\ell^{1}$ 和 $\ell^{2}$ 范数中找到与给定框架最近的平衡框架的问题，完整地表征了它的存在并给出了它的表达式。最后，我们提出了许多构建平衡单位范数紧框架的例子和方法。我们研究了在 $\ell^{1}$ 和 $\ell^{2}$ 范数中找到与给定框架最近的平衡框架的问题，完整地表征了它的存在并给出了它的表达式。最后，我们提出了许多构建平衡单位范数紧框架的例子和方法。我们研究了在 $\ell^{1}$ 和 $\ell^{2}$ 范数中找到与给定框架最近的平衡框架的问题，完整地表征了它的存在并给出了它的表达式。最后，我们提出了许多构建平衡单位范数紧框架的例子和方法。