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.
Categories: math.AP