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

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) (m)Φ and their Hankel transforms, the former ones being valid when m ∈ ℤ (+) is restricted to a suitable interval for which S (μ) (m)Φ 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/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).