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On a Hermite Interpolation by Polynomials of Two Variables

A problem of Hermite interpolation by polynomials of two variables is studied. The interpolation matches preassigned data of function values and consecutive normal derivatives on a set of points on several circles centered at the origin. It includes Lagrange interpolation as a special case. The uniqueness of the interpolation is established when the points are equidistant on the circles, while the points on different circles may differ by arbitrary rotations. This leads to a cubature formula on the unit disc, which can be given explicitly without knowing the explicit formula of the interpolation polynomial.