Interpolation by Hankel Translates of a Basis Function: Inversion Formulas and Polynomial Bounds. (29th October 2013) - Arteaga, Cristian; Marrero, Isabel
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 Y n (n ∈ ℕ ) depending on a weight w . The functions Φ and w are connected through the distributional identity t 4 n ( 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 m th 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 4 n 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 .