General Integral Transforms by the Concept of Generalized Reproducing Kernels

Abstract

In Saitoh (Proc Am Math Soc 89:74–78, 1983), the general integral transforms in the framework of Hilbert spaces were combined with the general theory of reproducing kernels by Aronszajn (Trans Am Math Soc 68:337–404, 1950) and many applications were developped, for example, in Saitoh (Integral transforms, reproducing kernels and their applications, vol 369, Addison Wesley Longman, Harlow, 1997). The basic assumption here that the integral kernels belong to some Hilbert spaces. However, as a very typical integral transform, in the case of Fourier integral transform, the integral kernel does not belong to L 2(R) and, however, we can establish the isometric identity and inversion formula.On the above situations, we will develop some general integral transform theory containing the Fourier integral transform case that the integral kernel does not belong to any Hilbert space, based on the recent general concept of generalized reproducing kernels in Saitoh and Sawano (Generalized delta functions as generalized reproducing kernels, manuscript; General initial value problems using eigenfunctions and reproducing kernels, manuscript).