A new statistical method is proposed for reduction of truncation artifacts when reconstructing a function by a finite number of its Fourier series coefficients. Following the Bayesian approach, it is possible to take into account both the errors induced by the truncation of the Fourier series and some specific characteristics of the function. A suitable Markov random field is used for modeling these characteristics. Furthermore, in applications like Magnetic Resonance Imaging, where these coefficients are the measured data, the experimental random noise in the data can also be taken into account. Monte Carlo Markov chain methods are used to make statistical inference. Parameter selection in the Bayesian model is also addressed and a solution for selecting the parameters automatically is proposed. The method is applied successfully to both simulated and real magnetic resonance images.