The relation between the Eulerian and the Lagrangian correlation functions is studied in two spatial dimensions. Simple analytical expressions for the full space-time varying Eulerian correlation are derived solely on the basis of the one-dimensional wavenumber power spectrum of the velocity fluctuations. It is demonstrated that an extension of the arguments giving the foregoing results allows also for derivation of analytical expressions for the Lagrangian autocorrelation function. The results are supported by direct numerical solutions of the non-dissipative Euler equations for the fluctuating velocity. The calculations are initialized by a flow with equilibrium spectral energy distributions. The Lagrangian correlation function is obtained by tracing a large number of passively convected test particles.