On some properties of the projection operator for a class of stabilization algorithms

The projection operator $Q[a]$ acting from the linear space of the functions $a (x) \in \mathrm{span} \{\sin i x,\; i \ge 1\}$ given on the segment $[0,\pi]$ onto the subspace of the functions $\tilde a(x) \in \mathrm{span} \{\sin i x,\; i > i_0\}$ is studied theoretically and numerically. The corresponding projection is performed along the subspace of the functions $l(x) \in \mathrm{span} \{{ \overline{\mathrm{ sin}}}\ i x , \; i=1,\ldots, i_0\}$, where ${ \overline{\mathrm{sin}}}\ i x = \chi_\delta (x) \sin i x$ is the characteristic function $\chi_{\delta} (x)$ of the interval $[0,\delta)$. The obtained results are used to solve the problem of stabilization with respect to the initial data of solutions to the model nonstationary equations.