Abstract

This M.Sc. thesis treats the feasibility of the Radon Split for solving radial integral equation involving radially acting integral operator on the Hardy-Lebesgue Class [Special characters omitted.] of the half-upper plane [Special characters omitted.] . In this process, we take a scrutinizing look at [Special characters omitted.] by means of the conformal map [Special characters omitted.] taking [Special characters omitted.] . We demonstrate that [Special characters omitted.] -functions f always possess a.e. unique boundary values [Special characters omitted.] [Special characters omitted.] as [Special characters omitted.] from within [Special characters omitted.] . These boundary values are also angular limit functions in the [Special characters omitted.] -sense--i.e. [Special characters omitted.] as [Special characters omitted.] from within [Special characters omitted.] . Concomitantly, the [Special characters omitted.] -parameter family of [Special characters omitted.] -kernels [Special characters omitted.] with uniformly bounded double norms, have unique angular limit [Special characters omitted.] -kernels [Special characters omitted.] in the [Special characters omitted.] -sense--i.e. [Special characters omitted.] = [Special characters omitted.] [arrow right] 0 as [Special characters omitted.] from within [Special characters omitted.] . These properties are consequences of the inverse Mellin-Transform, which transformation originates in Fourier-Plancherel Theorem for [Special characters omitted.] and [Special characters omitted.] . Because of this Mellin-Transform representation of [Special characters omitted.] and [Special characters omitted.] we may regard [Special characters omitted.] as the three entities: [Special characters omitted.] and [Special characters omitted.] , where the first two are Hilbert spaces and the third is a dual system with [Special characters omitted.] Consequently, we look upon [Special characters omitted.] as the Banach algebra [Special characters omitted.] and further as the Hilbert space [Special characters omitted.] . We successfully construct for every radial linear integral operator K of finite rank on [Special characters omitted.] , its transpose K T in [Special characters omitted.] as well as its adjoint K * in [Special characters omitted.] , which leans heavily on the interaction of * and T in [Special characters omitted.] . We prove a necessary and sufficient condition as to when an element of [Special characters omitted.] is radially representable. And finally, we construct Fredholm Resolvents not only finite-dimensional [Special characters omitted.] but also, by means of the Radon Split, the Fredholm Resolvents of any [Special characters omitted.] and that of its transpose K T in terms of [Special characters omitted.] . Herein, the Fredholm Alternatives are induced by the derivations.