Cubature formulas of Hermite type in the space of periodic functions of two variables
Abstract
In the present work, in the Sobolev space L2(m)˜(H), the lattice optimal cubature formulas of the Hermite type are constructed for periodic functions of two variables. Here the extremal function of the error functional of cubature formulas is found. Using the extremal function, the squared norm of the error functional of the considered lattice cubature formulas of the Hermite type is calculated. By minimizing the obtained norm, provided the cubature formula for the constant is accurate in terms of the coefficients, a system of linear equations is obtained in the form of equations in convolutions of functions of a discrete argument. In addition, the generalization of Babuška’s theorem is proved. Finally, the square of the norm of the error functional for the constructed optimal lattice cubature formulas is calculated.