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

摘要

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 Y n (n ∈ ℕ) depending on a weight w. The functions Φ and w are connected through the distributional identity t ⁴ⁿ(h μ′Φ)(t) = 1/w(t), where h μ′ 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 ℋ μ in order to derive explicit representations of the derivatives S μ ^mΦ and their Hankel transforms, the former ones being valid when m ∈ ℤ + is restricted to a suitable interval for which S μ ^mΦ is continuous. Here, S μ ^m denotes the mth iterate of the Bessel differential operator S μ if m ∈ ℕ, while S μ ⁰ is the identity operator. These formulas, which can be regarded as inverses of generalizations of the equation (h μ′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.