This paper presents a novel time-adaptive estimation technique by revisiting the classical Wiener-Hopf equation. Any convex and not necessarily differentiable function can be used for enlarging the Wiener-Hopf equation in order to incorporate the often met, in practice, measurement and model inaccuracies. Unlike classical techniques, e.g., the Recursive Least Squares (RLS) algorithm, the proposed method is free of the computation of the inverse of a correlation matrix. Moreover, the method offers the means for dealing with the presence of convex constraints in an efficient way, by exploiting general convex analytic tools. To validate the proposed estimation method, an application of increasing importance nowadays, the online sparse signal recovery task is considered. Numerical results support the introduced theoretical arguments against the sparsity-aware classical batch, and the very recently introduced RLS-based signal recovery techniques.