Summary: For a piecewise linear intermittent map, the evolution of statistical averages of a class of observables with respect to piecewise constant initial densities is investigated and generalized eigenfunctions of the Frobenius-Perron operator \(\widehat P\) are explicitly derived. The evolution of the averages are shown to be a superposition of the contributions from two simple eigenvalues 1 and \(\lambda_d\in(-1,0)\), and a continuous spectrum on the unit interval \([0,1]\) of \(\widehat P\). Power-law decay of correlations are controlled by the continuous spectrum. Also the non-normalizable invariant measure in the non-stationary regime is shown to determine the strength of the power-law decay.