Summary: A wavelet counterpart to Wahba’s spline smoothing technique is developed for the case of fitting less regular curves. Our approach is a realization of the smoothness penalty method of Tikhonov regularization of ill-posed inverse stochastic problems within a wavelet setting. The penalized cost functional to be minimized is Peetre’s \(K\)-functional between Besov spaces. The regularity of the curve is discussed in terms of the size of its seminorm in homogeneous Besov spaces \(\dot B_{pq}^s\) with a relatively small value of the smoothness index \(s>0\). Penalized \(L_2\)-estimation with Besov-type constraints, considered in the literature, is included as a partial case. The optimal solution of the penalization problem is in the form of a wavlet expansion whose coefficients are obtained by appropriate level- and/or space-dependent shrinking of the empirical wavelet coefficients. Thanks to the use of wavelets, both density and regression estimation can be treated in a somehow unified way. In the case of regression-function estimation, an enhanced version of cross validation (generalized full cross validation) is implemented for the choice of the smoothing parameter. Numerical examples illustrate the advantage of our procedures in comparison to more standard wavelet methods when the regularity is small and sample sizes are moderate. The approach is very flexible and allows for a diversity of extensions. Some first extensions can be found in Section 7, among which are the iterative individual shrinking estimator and the self-similar fractal estimator. The other extensions considered are grouped in Appendix B, which is in a sense the most advanced part of the paper, and can be considered as an outline of a plan for further research. For conciseness of presentation, only the univariate case is considered in detail, but all statements and their proofs do admit multidimensional generalizations.