arXiv 2402.094852024-02-14math.FA

TM系统能否构成Banach空间的无条件基？

Can TM system form an unconditional basis for Banach spaces?

Haibo Yang, Chitin Hon, Qixiang Yang, Tao Qian

AI总结：

本文解决了TM系统在Banach空间缺乏无条件基的问题，证明了两个TM系统能够构成 $\mathbb{H}^{p}(D)$ 的无条件基，为其扩展至再生核Banach空间算法提供了理论基础。

摘要：

解析信号算法的研究长期以来备受关注。Takenaka-Malmquist (TM)系统于1925年被引入以研究解析函数。如果TM系统满足双曲不可分条件，则它是一个正交基。它可以构成Hilbert空间 $\mathbb{H}^{2}(D)$ 的无条件基和Banach空间 $\mathbb{H}^{p}(D)(1 < p < \infty)$ 的Schauder基。在刻画函数空间时，一个必要条件是该基是否为无条件的。但自1925年引入TM系统以来，据我们所知，尚未有人证明存在能够构成Banach空间 $\mathbb{H}^{p}(D) (p \neq 2)$ 无条件基的TM系统。TM系统具有简单直观的解析结构，因此也被应用于学习算法，并系统地发展到再生核Hilbert空间 (RKHS)。由于缺乏无条件基性质，它无法扩展到再生核Banach空间 (RKBS) 算法。但Banach空间的情况在机器学习中起着重要作用。本文证明了两个TM系统可以构成 $\mathbb{H}^{p}(D) (1<p <\infty)$ 的无条件基。

英文摘要：

The research on the algorithm of analytic signal has received much attention for a long time. Takenaka-Malmquist (TM) system was introduced to consider analytic functions in 1925. If TM system satisfies hyperbolic inseparability condition, then it is an orthogonal basis. It can form unconditional basis for Hilbert space $\mathbb{H}^{2}(D)$ and Schauder basis for Banach space $\mathbb{H}^{p}(D)(1 < p < \infty)$. In characterizing a function space, a necessary condition is whether the basis is unconditional. But since the introduction of TM systems in 1925, to the best of our knowledge, no one has proved the existence of a TM system capable of forming an unconditional basis for Banach space $\mathbb{H}^{p}(D) (p \neq 2)$. TM system has a simple and intuitive analytical structure. Hence it is applied also to the learning algorithms and systematically developed to the reproducing kernel Hilbert spaces (RKHS). Due to the lack of unconditional basis properties, it cannot be extended to the reproducing kernel Banach spaces (RKBS) algorithm. But the case of Banach space plays an important role in machine learning. In this paper, we prove that two TM systems can form unconditional basis for $\mathbb{H}^{p}(D) (1<p <\infty)$.

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