Weight optimal cubature formulas in Sobolev's periodic space

In the present paper the weight lattice optimal cubature formulas in the periodic Sobolev's space $\widetilde L_2^{(m)}(\Omega)$ are constructed. Under writing out of the algorithm of the construction the extremal function is found, and by means of the function the norm of the error functional of the cubature formula is calculated. Minimizing the norms, periodic Winier–Hopf's systems are obtained. Then the uniqueness solution to the system has been proved.