Hyperbolic Fourier series

Abstract

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 $1$, $\exp (i\pi nx)$, $\exp (i\pi n/x)$, $n\in \mathbb{Z}\setminus \{0\}$, and show that it is complete in $L^1(\mathbb{R})$. We associate with each $f\in L^1(\mathbb{R},(1+x^2{)}^{-1}dx)$ its hyperbolic Fourier series $h_0(f)+{\sum }_{n\in \mathbb{Z}\setminus \{0\}}(h_n(f)e^{i\pi nx}+m_n(f)e^{-i\pi n/x})$ 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 $U_{\phi }(x,y):={\int }_{\mathbb{R}}\phi (t)\exp \left(ixt+iy/t\right)dt$, for $\phi \in L^1(\mathbb{R})$ and $(x,y)\in {\mathbb{R}}^2$, supplies interpolating functions for the Klein-Gordon equation.

123 pages, 3 figures

References in corpus (2)

• The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps
• Exponential integral representations of theta functions