Best approximations of the classes B p,θ r of periodic functions of many variables in uniform metric

We obtain estimates exact in order for the best approximations of the classes B ∞,θ r of periodic functions of two variables in the metric of L ∞ by trigonometric polynomials whose spectrum belongs to a hyperbolic cross. We also investigate the best approximations of the classes B p,θ r , 1 ≤ p < ∞, of periodic functions of many variables in the metric of L ∞ by trigonometric polynomials whose spectrum belongs to a graded hyperbolic cross.