On the Autoconvolution Equation and Total Variation Constraints

This paper is concerned with the numerical analysis of the autoconvolution equation x * = = y icted to the interval [0, 1]. We present a discrete constrained least squares approach and prove its convergence in L p (0, 1), 1 ≤ p < ∞, where the regularization is based on a prescribed bound for the total variation of admissible solutions. This approach includes the case of non-smooth solutions possessing jumps. Moreover, an adaptation to the Sobolev space H 1 (0, 1) is added. A numerical case study concerning the determination of non-monotone smooth and non-smooth functions x from the autoconvolution equation with noisy data y completes the paper.