AI 中文总结
该研究证明了紧群上希尔伯特空间值函数的佩戈定理，有限维时等价关系成立，无限维时引入一致紧性条件可恢复等价关系，还建立了相关向量值傅里叶变换的核心不等式。
AI 中文摘要
我们证明了紧群上希尔伯特空间值函数的佩戈紧性定理的类似结果。对于取值于希尔伯特空间而非复数的平方可积函数，我们证明：一个有界族是预紧的，当且仅当它同时在两个互补意义上表现良好——其成员在群的小平移下变化不大，且它们的傅里叶系数在该族上一致衰减。当希尔伯特空间是有限维时，该等价关系无条件成立；当希尔伯特空间仅为复数空间时，该等价关系退化为已知的标量值定理。随后我们构造了一个具体例子，表明一旦允许希尔伯特空间为无限维，该等价关系就会真正失效。为修正这一问题，我们引入了一致紧性条件，并证明在该额外假设下，无论希尔伯特空间的维数如何，该等价关系均可恢复。在此过程中，我们建立了该向量值傅里叶变换的普朗谢尔等距、豪斯多夫-杨不等式及其逆不等式。
英文摘要
We prove a Hilbert space-valued analogue of Pego's compactness theorem on compact groups. For square-integrable functions taking values in a Hilbert space rather than in the complex numbers, we show that a bounded family is precompact exactly when it is simultaneously well behaved in two complementary senses: its members do not change much under small translations of the group, and their Fourier coefficients decay uniformly across the family. This equivalence holds without restriction when the Hilbert space is finite-dimensional, and it specializes to the known scalar-valued theorem when the Hilbert space is just the complex numbers. We then construct an explicit example showing that the equivalence genuinely breaks down once the Hilbert space is allowed to be infinite-dimensional. To repair this, we introduce a uniform tightness condition and we show that under this extra hypothesis the equivalence is restored regardless of the dimension of the Hilbert space. Along the way we establish the Plancherel isometry, the Hausdorff-Young inequality and its inverse for this vector-valued Fourier transform.
CommentsIn memory of Professor Koffi Kenny Siggini
