McKennon-Type Spaces for the Hankel Transformation

Forμ>−\frac12;, the Zemanian space Hμof Hankel-transformable functions is expressed as the projective limit of a chain {Spμ}p∈Nof reflexive Banach spaces where the Hankel transformation Sμis an isometric isomorphism. Moreover, Hμis dense in Spμ(p∈N) and Spμis dense in Sqμ(p, q, ∈N, p⩾q). The images of these results in the dual spaces then follow. A fundamental rôle in the construction is played by the space S0μ, consisting of all thoseφ=φ(t) (t∈I=]0, ∞[) such that bothφand Sμφlie in the spaceLμof functions Lebesgue integrable onIwith respect to the weighttμ+1/2. We show that, under Hankel convolution, S0μis a Banach algebra and a dense ideal ofLμ, namely that formed by the functions inLμwhich are Hankel–Stieltjes transforms of Radon measuresγonIwith the property that ∫∞0 tμ+1/2d |γ|(t)<∞. New results on the inversion of the Hankel transformation and the positivity of certain Hankel transforms are also established.