框架、半框架和希尔伯特尺度
Frames, semi-frames, and Hilbert scales
摘要：
给定希尔伯特空间中的一个全序列，我们称之为上（或上）序列。下）半框架，如果只有上（分别下）帧边界是有效的。等价地，对于上半框架，框架算子是有界的，但有一个无界的逆，而下半框架有一个无界的框架算子，有界的逆。对于上半帧，在离散和连续的情况下，我们建立了两个自然的希尔伯特尺度，这可能会产生一个新的表征某些功能空间的信号处理的兴趣。我们提出了一些例子，此外，一些结果关于下半框架和上半框架之间的对偶，以及一些推广，包括融合半框架和Banach半框架。
英文摘要：
Given a total sequence in a Hilbert space, we speak of an upper (resp. lower) semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently, for an upper semi-frame, the frame operator is bounded, but has an unbounded inverse, whereas a lower semi-frame has an unbounded frame operator, with bounded inverse. For upper semi-frames, in the discrete and the continuous case, we build two natural Hilbert scales which may yield a novel characterization of certain function spaces of interest in signal processing. We present some examples and, in addition, some results concerning the duality between lower and upper semi-frames, as well as some generalizations, including fusion semi-frames and Banach semi-frames.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
