= Exponentially coupled Ising inverse kernel
{title2=$J^{-1}(q)=(\cosh\kappa-\cos q)/(J\sinh\kappa)$}
For the infinite one-dimensional coupling <matrix> $J_{ij}=Je^{-\kappa|i-j|}$ with $J,\kappa>0$, its Fourier symbol is $J\sinh\kappa/(\cosh\kappa-\cos q)$. Its inverse is nearest-neighbor even though the original interaction is not. The inverse quadratic form equals $[2J\sinh\kappa]^{-1}\sum_j(m_j-m_{j+1})^2+[\tanh(\kappa/2)/J]\sum_jm_j^2$. Finite open chains add endpoint corrections; a periodized coupling gives the same translation-invariant formula.
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