A kernel theorem from the Hankel transform in Banach spaces

Abstract

One of the cornerstones in distribution theory was the kernel theorem of Schwartz in 1957, which showed that every bilinear continuous functional f(ϕ, ψ) on the space D(Ω1)×D(Ω2) can be represented by a linear continuous functional g on the space D(Ω1×Ω2). Zemanian [Zemanian, A.H., 1972, Realizability Theory for Continuous Linear Systems (New York: Academic Press).] extended the theorem to a more general type of product space where V is a Fréchet space. His work was based on the fact that the space is an inductive limit space and the convolution product is well defined in . In this paper, we study a new product space H μ×A, where H μ is the testing space for the classical Hankel transform and A is a Banach space, and derive the kernel theorem which is considered as a unified form for integral transforms such as Mellin, Laplace, Hankel and the K-transform by choosing particular Banach spaces for A. Using the Hankel transform of arbitrary order and pseudo-integrals, we find a generalized solution in H μ′ for the following differential equation: where J μ(x) is the Bessel function of first kind and order μ≠−1,−2,−3, ….