 Exponential-kernel half-line eigenfunction

The equation ${\int }_0^{\infty }{e}^{-|x-t|}f(t)dt=f(x)$ has this solution family in the ordinary bounded-growth transform class. Differentiating twice gives $f^{\prime \prime }=-f$, and the integral's right derivative at zero gives $f^{\prime }(0)=f(0)$. Alternatively its positive-support Fourier transform is $A(ik-1)/(k^2-1)$; inversion above both real poles gives the same answer.