On Practical Asymptotically Optimal Cubature Formulas in Sobolev Space \(\bar{L}_{p}^{\left( m \right)}\left( {{S}_{n}} \right)\)

In investigating various problems related to the theory of approximate integration, partial differential equations, and other areas of analysis, the functional approach has proven to be highly effective. This approach involves treating the differential equation with boundary conditions as an operator acting in a specifically chosen functional space. The necessary information is derived from the properties of this operator. When addressing problems in approximate integration and differential equations, the appropriate selection of functional spaces is crucial for success. S.L. Sobolev exemplified this method clearly in his well-known study of polyharmonic equations. In our research, we examined cubature formulas within the Sobolev functional space \(\bar{L}_{p}^{\left( m \right)}\left( {{S}_{n}} \right)\) for functions defined on \(n\)-dimensional unit sphere \(S\). This issue requires careful attention when developing the most efficient formulas. In this paper, we discussed formulas that fulfill this criterion and refered to them as “practical” in accordance with N.S. Bakhvalov’s terminology.

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