Abstract
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 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.
