We obtain order estimates for the best uniform orthogonal trigonometric approximations of $2\pi$-periodic functions, whose $(\psi,\beta)$-derivatives belong to unit balls of spaces $L_{ p}, \ 1\leq p<\infty$, in case at consequences $\psi(k)$ are that product $\psi(n)n^{ \frac{ 1}{ p}}$ can tend to zero slower than any power function and $\sum\limits_{ k=1}^{ \infty}\psi^{ p'}(k)k^{ p'-2}<\infty$ when $1<p<\infty$, $\frac{ 1}{ p}+\frac{ 1}{ p'}=1$ and $\sum\limits_{ k=1}^{ \infty}\psi(k)<\infty$ when $p=1$. We also establish the analogical estimates in $L_{ s}$-metric, $1< s\leq \infty$, for classes of the summable $(\psi,\beta)$-differentiable functions, such that $\parallel f_{ \beta}^{ \psi}\parallel_{ 1}\leq1$.
Categories: math.CA