Hyperbolic Fourier series and the Klein-Gordon equation

• DOI: 10.48550/arxiv.2401.06871
• Publication Date: 2024-01-01
• Authors: Andrew Bakan; Håakan Hedenmalm; Alfonso Montes-Ro...; Danylo Radchenko; M. Viazovska

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