We discuss the problem of adaptive discrete-time signal denoising in the situation where the signal to be recovered admits a ``linear oracle'' - an unknown linear estimate that takes the form of convolution of observations with a time-invariant filter. It was shown by Juditsky and Nemirovski (2009) that when the $\ell_2$-norm of the oracle filter is small enough, such oracle can be ``mimicked'' by an efficiently computable \textit{adaptive} estimate of the same structure with the observation-driven filter. The filter in question was obtained as a solution to the optimization problem in which the $\ell_\infty$-norm of the Discrete Fourier Transform (DFT) of the estimation residual is minimized under constraint on the $\ell_1$-norm of the filter DFT. In this paper, we discuss a new family of adaptive estimates which rely upon minimizing the $\ell_2$-norm of the estimation residual. We show that such estimators possess better statistical properties than those based on~$\ell_\infty$-fit; in particular, under the assumption of \textit{approximate shift-invariance} we prove oracle inequalities for their $\ell_2$-loss and improved bounds for $\ell_2$- and pointwise losses. We also study the relationship of the approximate shift-invariance assumption with the signal simplicity introduced by Juditsky and Nemirovski (2009), and discuss the application of the proposed approach to harmonic oscillation denoising.