On the best simultaneous “angle” approximation in the mean of periodic functions of two variables from some classes

In the $L_2$ metric, we obtain sharp inequalities between the best joint approximations of $2\pi$-periodic functions $f(x,y)$ differentiable in each of the variables and their successive derivatives $f^{(\mu,\nu)}( x,y) \ (\mu=0,1,\ldots,r; \nu=0,1,\ldots,s)$ by trigonometric “angles” with double integrals containing mixed moduli of continuity of higher orders of higher derivatives. The sharp values of the upper bound of the best joint approximation of some classes of functions given by the specified moduli of continuity are found.