For now wavelet estimation is an important part of the nonparametric curve estimation. The minimax theory is well developed and there are many interesting practical applications where wavelets perfectly fit purposes of the data analysis. This paper presents a new theoretical result and a new application. It is well known in the statistical wavelet theory that for a Besov space of order the minimax rate of the mean integrated squared error (MISE) convergence is . The developed in the paper local minimax approach shows that this rate is too slow and that any Besov’s function of order can be estimated with a faster rate. Furthermore, an adaptive estimator is proposed that is minimax under the classical and new minimax approaches. Then an interesting and challenging application of the adaptive wavelet estimator in the fMRI study of neuroplasticity is discussed.