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

Abstract

For μ≥−1/2, the authors have developed elsewhere a scheme for interpolation by Hankel translates of a basis function Φ in certain spaces of continuous functions Yn (n∈ℕ) depending on a weight w. The functions Φ and w are connected through the distributional identity t4n(hμ′Φ)(t)=1/w(t), where hμ′ denotes the generalized Hankel transform of order μ. In this paper, we use the projection operators associated with an appropriate direct sum decomposition of the Zemanian space ℋμ in order to derive explicit representations of the derivatives SμmΦ and their Hankel transforms, the former ones being valid when m∈ℤ+ is restricted to a suitable interval for which SμmΦ is continuous. Here, Sμm denotes the mth iterate of the Bessel differential operator Sμ if m∈ℕ, while Sμ0 is the identity operator. These formulas, which can be regarded as inverses of generalizations of the equation (hμ′Φ)(t)=1/t4nw(t), will allow us to get some polynomial bounds for such derivatives. Corresponding results are obtained for the members of the interpolation space Yn.