Order estimates of best orthogonal trigonometric approximations of classes of infinitely differentiable functions

In this paper we establish exact order estimates for the best uniform orthogonal trigonometric approximations of the classes of $2\pi$-periodic functions, whose $(\psi ,\beta )$-derivatives belong to unit balls of spaces $L_p$, $1\le p<\mathrm{\infty }$, in the case, when the sequence $\psi (k)$ tends to zero faster, than any power function, but slower than geometric progression. Similar estimates are also established in the $L_s$-metric, $1<s\le \mathrm{\infty }$ for the classes of differentiable functions, which $(\psi ,\beta )$-derivatives belong to unit ball of space $L_1$.