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The a priori evaluation of the pointwise approximation, by a regu- larization method suggested earlier, of a L1 periodic function f, when only noisy Fourier coefficients of f are known, is given here in a formulation more suitable for numerical verification. A careful evaluation of the constants in- volved in the formulas is also provided. Finally, a procedure is described of a statistical check performed in order to verify whether the theoretically suggestedvalue of the regularization parameter is a good choice with respect to functionsgenerally arising in applications. 1. IntroductionWe consider the (generally) ill-posed problem of "reconstructing" a function/, integrable on the A-dimensional torus, when we only know the sequence ofits Fourier coefficients. A recent paper [2] (see also [1]) describes a regularizationmethod to solve this problem which is stable also in the case of noisy data. Extensive numerical experience shows that the method is very efficient [4] (seealso [5]). The efficiency is closely related to the choice of the regularizationparameter cr. In [2], for large classes of functions, evaluations are given of the