Cubature formulas that are exact for trigonometric polynomials

Under more general assumptions about the integral, the known lower bound for the number of nodes of a cubature formula is obtained that is exact for trigonometric polynomials of degree no higher than 2k + 1. A theorem is proved that provides the necessary and sufficient condition for the existence of a special type of cubature formula that is exact for trigonometric polynomials of degree no higher than 2k + 1 and has the number of nodes equal to the lower bound.