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

Abstract

Forμ≥−1/2, the authors have developed elsewhere a scheme for interpolation by Hankel translates of a basis functionΦin certain spaces of continuous functionsYn(n∈ℕ) depending on a weightw. The functionsΦandware connected through the distributional identityt4n(hμ′Φ)(t)=1/w(t), wherehμ′denotes the generalized Hankel transform of orderμ. In this paper, we use the projection operators associated with an appropriate direct sum decomposition of the Zemanian spaceℋμin order to derive explicit representations of the derivativesSμmΦand their Hankel transforms, the former ones being valid whenm∈ℤ+is restricted to a suitable interval for whichSμmΦis continuous. Here,Sμmdenotes themth iterate of the Bessel differential operatorSμifm∈ℕ, whileSμ0is 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 spaceYn.