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We show that wavelets can simultaneously achieve the regularization of an ill-posed problem such as the extrapolation of a band-limited signal and provide the discretized solution. The signal is supposed to be known only on the subset Aλ = λA, where λ is a large-scale parameter and where A⊂ℝd is a compact convex body. When the Fourier transform of the signal has a compact convex support, we show, using the Malvar-Wilson wavelets, how to regularize the equation Kf = g where K is the composition of the band-limiting and time-limiting operators. Results of Slepian, Landau and Al for the SVD techniques are retrieved: wavelets are an efficient way of regularizing when the singular value computations are difficult. We also propose an iterative scheme based on a linear system to solve the above problem. In fact, we shall see that the matrix involved to solve the problem using our wavelets is almost diagonal. Finally, some numerical results involving uni-dimensional and bi-dimensional signals will be exhibited to assess the performance of our technique.

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