# Hyperbolic Fourier series

Andrew Bakan, Haakan Hedenmalm, Alfonso Montes-Rodriguez, Danylo Radchenko, Maryna Viazovska

Andrew Bakan, Haakan Hedenmalm, Alfonso Montes-Rodriguez, Danylo Radchenko, Maryna Viazovska

**1/10/21**Published in : arXiv:2110.00148

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.