Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables

We obtained the exact order estimates of the best orthogonal trigonometric approximations of periodic functions of one and several variables from the Nikol'skii-Besov-type classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ in the multivariate case $d\geq2$) in the space $B_{\infty,1}$. We observe that in the multivariate case the orders of mentioned approximation characteristics of the functional classes $B^{\Omega}_{1,\theta}$ are realized by their approximations by step hyperbolic Fourier sums that contain the necessary number of harmonics. In the univariate case, an optimal in the sense of order estimates for the best orthogonal trigonometric approximations of the corresponding functional classes are the ordinary partial sums of their Fourier series. As a consequence of the obtained results, the exact order estimates of the orthowidths of the classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ for $d\geq2$) in the space $B_{\infty,1}$ are also established. Besides, we note that in the univariate case, in contrast to the multivariate one, the estimates of the considered approximation characteristics do not depend on the parameter $\theta$.