Interpolation by Hankel Translates of a Basis Function: Inversion Formulas and Polynomial Bounds

For mu >= -1/2, the authors have developed elsewhere a scheme for interpolation by Hankel translates of a basis function Phi in certain spaces of continuous functions Y-n (n is an element of N) depending on a weight w. The functions Phi and w are connected through the distributional identity t(4n)(h'(mu)Phi)(t) = 1/w(t), where h'(mu) denotes the generalized Hankel transform of order mu. In this paper, we use the projection operators associated with an appropriate direct sum decomposition of the Zemanian space H-mu in order to derive explicit representations of the derivatives S-mu(m)Phi and their Hankel transforms, the former ones being valid when m is an element of Z(+) is restricted to a suitable interval for which S-mu(m)Phi is continuous. Here, S-mu(m) denotes the mth iterate of the Bessel differential operator S-mu if m is an element of N, while S-mu(0) is the identity operator. These formulas, which can be regarded as inverses of generalizations of the equation (h'(mu)Phi)(t) = 1/t(4n)w(t), will allow us to get some polynomial bounds for such derivatives. Corresponding results are obtained for the members of the interpolation space Y-n.