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- W3204549188 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 $1$, $exp ( i pi n x)$, $exp ( i pi n/ x)$, $n in mathbb{Z} setminus {0}$, 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." @default.
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- W3204549188 date "2021-10-01" @default.
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- W3204549188 title "Hyperbolic Fourier series." @default.
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