  In this article we explain the essence of the interrelation described in
[PNAS 118, 15 (2021)] on how to write explicit interpolation formula for
solutions of the Klein-Gordon equation by using the recent Fourier pair
interpolation formula of Viazovska and Radchenko from [Publ Math-Paris 129, 1
(2019)]. We construct explicitly the sequence in $L^1 (\mathbb{R} )$ which is
biorthogonal to the system $1$, $\exp ( i \pi n x)$, $\exp ( i \pi n/ x)$, $n
\in \mathbb{Z} \setminus \{0\}$, and show that it is complete in $L^1
(\mathbb{R})$. We associate with each $f \in L^1 (\mathbb{R}, (1+x^2)^{-1} d
x)$ its hyperbolic Fourier series
  $h_{0}(f) + \sum_{n \in \mathbb{Z}\setminus \{0\}}(h_{n}(f) e^{ i \pi n x} +
m_{n}(f) e^{-i \pi n / x} )$ and prove that it converges to $f$ in the space of
tempered distributions on the real line. Applied to the above mentioned
biorthogonal system, the integral transform given by
  $U_{\varphi} (x, y):= \int_{\mathbb{R}} \varphi (t) \exp \left( i x t + i y /
t \right) d t $, for $\varphi \in L^{1} (\mathbb{R})$ and $(x, y) \in
\mathbb{R}^{2}$, supplies interpolating functions for the Klein-Gordon
equation.
