Numerical integration over a disc. A new Gaussian quadrature formula

The ordinary type of information data for approximation of functions f or functionals of them in the univariate case consists of function values {f(x1), . . . , f(xm)}. The classical Lagrange interpolation formula and the Gauss quadrature formula are famous examples. The simplicity of the approximation rules, their universality, the elegancy of the proofs and the beauty of these classical results show that the function values are really the most natural pieces of information in the reconstruction of functions and functionals. The direct transformation of the univariate results to the multivariate setting faces however various difficulties. For example, the problem of constructing a polynomial P (x, y) of degree n,