On the construction of practically asymptotically optimal weighted cubature formulas of Hermite type in the space of S.L. Sobolev $\bar{L}_{2}^{\left( m \right)}\left( {{S}_ {n}}\right)$

When solving many problems in the theory of approximate integration and differential equations, it is the correct choice of spaces that is the key to success. A very clearly chosen approach was demonstrated in the famous works of S.L. Sobolev on the polyharmonic equation. S.L. Sobolev posed and solved by the variational method the first boundary value problem for the equation ${{\Delta }^{\ell }}u=f$ with boundary conditions on surfaces of various dimensions.
Problems of optimization of approximate integration formulas consist in minimizing the norm of the error functional of the formula on selected normalized spaces, and most of them are considered in the Sobolev space.
Until now, we have considered cubature formulas, with the help of which a definite integral of a function is approximately calculated when the values of this function at individual points of the nodes of the cubature formula are unknown. But more general cubature formulas are possible, which include both the values of the function and the values of its derivatives of one order or another.
If we know not only the values of a function at some points of the $n-$ dimensional unit sphere, but also the values of its derivatives of one order or another, then it is natural that if all this data is used correctly, we can expect a more accurate result than if we use only function values.
This paper examines cubature formulas, which require special attention to the construction of the most economical formulas; according to N.S. Bakhvalov, such formulas are called practical.