  Finite temperature Euclidean two-point functions in quantum mechanics or
quantum field theory are characterized by a discrete set of Fourier
coefficients $G_{k}$, $k\in\mathbb Z$, associated with the Matsubara
frequencies $\nu_{k}=2\pi k/\beta$. We show that analyticity implies that the
coefficients $G_{k}$ must satisfy an infinite number of model-independent
linear equations that we write down explicitly. In particular, we construct
"Analytic Renormalization Group" linear maps $\mathsf A_{\mu}$ which, for any
choice of cut-off $\mu$, allow to express the low energy Fourier coefficients
for $|\nu_{k}|<\mu$ (with the possible exception of the zero mode $G_{0}$),
together with the real-time correlators and spectral functions, in terms of the
high energy Fourier coefficients for $|\nu_{k}|\geq\mu$. Operating a simple
numerical algorithm, we show that the exact universal linear constraints on
$G_{k}$ can be used to systematically improve any random approximate data set
obtained, for example, from Monte-Carlo simulations. Our results are
illustrated on several explicit examples.
