Can TM system form an unconditional basis for Banach spaces?

Abstract

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 $\mathbb{H}^{p}(D) (1<p <\infty)$.