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

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.

The pseudodifferential operatorA(x,D)

The pseudodifferential operator (p.d.o.)A(x,D), associated with the Bessel operatord2/dx2+(1−4μ2)/4x2, is defined. Symbol classHρ,δmis introduced. It is shown that the p.d.o. associated with a symbol belonging to this class is a continuous linear mapping of the Zemanian spaceHμinto itself. An integral representation of p.d.o. is obtained. Using Hankel convolutionLσ,αp-norm continuity of the p.d.o. is proved.

Some Properties of Hankel Convolution Operators

Let be the Zemanian space of Hankel transformable generalized functions and let be the space of Hankel convolution operators for . This is the dual of a subspace of for which is also the space of Hankel convolutors. In this paper the elements of are characterized as those in and in that commute with Hankel translations. Moreover, necessary and sufficient conditions on the generalized Hankel transform are established in order that every such that in .