Two-Dimensional Minimal Cubature Formulas and Matrix Equations

For strictly positive, linear, and centrally symmetric functionals in two dimensions the existence of cubature formulas attaining the known lower bounds is equivalent to the solvability of certain matrix equations under some constraints. Any solution generates a real ideal the common roots of which are the nodes of the cubature formula. These results are applied to construct an infinite number of minimal positive cubature formulas of an arbitrary degree of exactness for one special, but classical, integral.

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