Finite Fourier self-transforms

Abstract

This paper considers the integral equation \[ λ γ ( t ′ ) = ( 2 π ) − N ∫ Ω exp ⁡ ( − i ω ⋅ t ′ ) ∫ T γ ( t ) exp ⁡ ( i ω ⋅ t ) d t d ω \lambda \gamma \left ( {t’} \right ) = {\left ( {2\pi } \right )^{ - N}}\int \limits _\Omega {\exp \left ( { - i\omega \cdot t’} \right )\int \limits _T {\gamma \left ( t \right )} \exp \left ( {i\omega \cdot t} \right )dt d\omega } \] as well as a more general one wherein the Fourier kernels are weighted. When Ω \Omega and T T are N N -dimensional spherical domains, the eigenfunctions of the integral equation are generalized prolate spheroidal functions for which a new nomenclature is proposed. Many properties of the eigenfunctions are developed and summarized. Because of the importance of these functions in Fourier transform theory, old as well as new properties are included.