Polynomial bases for subspaces of vector fields in the unit ball. Method of ridge functions

Published Online: 2007-05-31 This paper deals with the construction of orthogonal polynomial bases for particular subspaces of vector fields defined in the unit ball of ℝ^3. Our approach uses vector spherical harmonics to construct orthogonal sets of specific solenoidal and potential vector fields by means of ridge functions. It is shown that the approach leads to bases according to the subspaces induced by the Helmholtz–Hodge decomposition of square integrable vector fields. Key Words: vector field,; harmonic field,; spherical harmonics,; ridge function,; Helmholtz–Hodge decomposition,; Zernike polynomials,; Funk–Hecke theorem.