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