The complete C(N)D-solution of the problem recovery of functions from anisotropic Sobolev classes by their unexact trigonometric Fourier coefficients

In the case, when numerical information is removed from Fourier trigonometric coefficients - one of the basic linear functionals in the full volume is solved C(N)D – problem of optimal restoration of functions from $W_2^{r_1,\ldots,r_s}(0,1)^s$ anisotropic Sobolev classes. Exactly, are written $\delta _{N} (0;\, \Phi _{N} )_{L^{2} } \asymp \, N^{-(r_1^{-1} +...+r_s^{-1} )^{-1} } $ unimproved order of restoration of functions by their exact values of Fourier trigonometric coefficients. Based on this is constructed сomputing unit, which is trigonometric polynomial and is found amount $\tilde{\varepsilon }_{N} =N^{-(r_1^{-1} +...+r_s^{-1} )^{-1} -\frac{1}{2}}$ of error calculating Fourier coefficients, saving unimproved order of restoration by accurate information. Then is shown that сomputing unit by Fourier trigonometric coefficients with better error do not exist.