Hyperbolic Fourier series

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 L1(R) which is biorthogonal to the system 1, exp(iπnx), exp(iπn/x), n∈Z∖{0}, and show that it is complete in L1(R). We associate with each f∈L1(R,(1+x2)−1dx) its hyperbolic Fourier series h0(f)+∑n∈Z∖{0}(hn(f)eiπnx+mn(f)e−iπn/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 Uφ(x,y):=∫Rφ(t)exp(ixt+iy/t)dt, for φ∈L1(R) and (x,y)∈R2, supplies interpolating functions for the Klein-Gordon equation.