Cubature Rules for Harmonic Functions Based on Radon Projections

We construct a class of cubature formulae for harmonic functions on the unit disk based on line integrals over 2n + 1 distinct chords. These chords are assumed to have constant distance t to the center of the disk, and their angles to be equispaced over the interval [0, 2π]. If t is chosen properly, these formulae integrate exactly all harmonic polynomials of degree up to 4n + 1, which is the highest achievable degree of precision for this class of cubature formulae. For more generally distributed chords, we introduce a class of interpolatory cubature formulae which we show to coincide with the previous formulae for the equispaced case. We give an error estimate for a particular cubature rule from this class.