In an earlier paper the same authors [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 1, No. 4, 305-308 (1990; Zbl 0711.42014)] presented a general method for the reconstruction of a periodic function knowing only an approximation of its Fourier coefficients. Here, this method is widely illustrated and improved by adding also the proofs of the theorems stated in the above paper.