For the problem of nonparametric detection of signal in Gaussian white noise, strong asymptotically minimax tests are found. The sets of alternatives are balls in the Besov space B^s_2∞ with “small” balls in L₂ removed. The balls in the Besov space are defined in terms of orthogonal expansions of functions in trigonometrical basis. Similar result is also obtained for nonparametric hypothesis testing on a solution of ill-posed linear inverse problem with Gaussian random noise. Bibliography: 19 titles.