Hyperbolic Fourier series and the Klein-Gordon equation

Abstract

In an effort to extend classical Fourier theory, Hedenmalm and Montes-Rodr\'ıguez (2011) found that the function system \[ e_m(x)=e^{iπmx},\quad e_n^\dagger(x)=e_n(-1/x)=e^{-iπn/x} \] is weak-star complete in $L^{\infty }(\mathbb{R})$ when $m,n$ range over the integers with $n\ne 0$. It turns out that the system can be used to provide unique representation of functions and more generally distributions on the real line $\mathbb{R}$. For instance, we may represent uniquely the unit point mass at a point $x\in \mathbb{R}$: \[ δ_x(t)=A_0(x)+\sum_{n\ne0}\big(A_n(x)\,e^{iπnt} +B_n(x)\,e^{-iπn/t}\big), \] with at most polynomial growth of the coefficients, so that the sum converges in the sense of distribution theory. In a natural sense, the system $\{A_n,B_n{\}}_n$ is biorthogonal to the initial system $\{e_n,e_n^{†}{\}}_n$ on the real line. More generally, for a distribution $f$ on the compactified real line, we may decompose it in a \emph{hyperbolic Fourier series} \[ f(t)=a_0(f)+\sum_{n\ne0}\big(a_n(f)\,e^{iπnt}+b_n(f)\,e^{-iπn/t}\big), \] understood to converge in the sense of distribution theory. Such hyperbolic Fourier series arise from two different considerations. One is the Fourier interpolation problem of recovering a radial function $\varphi$ on ${\mathbb{R}}^d$ from partial information on $\varphi$ and its Fourier transform $\hat{\varphi }$, studied by Radchenko and Viazovska (2019). Another consideration is the interpolation theory of the Klein-Gordon equation ${\partial }_x{\partial }_{y}u+u=0$. For instance, the biorthogonal system $\{A_n,B_n{\}}_n$ leads to a collection of solutions that vanish along the lattice-cross of points $(\pi k,0)$ and $(0,\pi l)$ save for one of these points. These interpolating solutions allow for restoration of a given solution $u$ from its values on the lattice-cross.

71 pages