Pulsed gradient spin echo (PGSE) is a well-known NMR technique for determining diffusion coefficients. Various signal processing techniques have been introduced to solve the task, which is especially challenging when the decay is multiexponential with an unknown number of components. Here, we introduce a new method for the processing of such types of signals. Our approach modifies the Tikhonov's regularization, known previously in CONTIN and Maximum Entropy (MaxEnt) methods, by using the l(1)-norm penalty function. The modification enforces sparsity of the result, which improves resolution, compared to both mentioned methods. We implemented the Iterative Thresholding Algorithm for Multiexponential Decay (ITAMeD), which employs the l(1)-norm minimization, using the Fast Iterative Shrinkage Thresholding Algorithm (FISTA). The proposed method is compared with the Levenberg-Marquardt-Fletcher fitting, Non-negative Least Squares (NNLS), CONTIN, and MaxEnt methods on simulated datasets, with regard to noise vulnerability and resolution. Also, the comparison with MaxEnt is presented for the experimental data of polyethylene glycol (PEG) polymer solutions and mixtures of these with various molecular weights (1080 g/mol, 11,840 g/mol, 124,700 g/mol). The results suggest that ITAMeD may be the method of choice for monodispersed samples with "discrete" distributions of diffusion coefficients.