Novel sampling formulas associated with quaternionic prolate spheroidal wave functions

TL;DR

This paper introduces new sampling formulas for quaternion-valued signals using quaternion reproducing kernel Hilbert spaces, extending classical sampling theorems and linking to prolate spheroidal wave functions.

Contribution

It develops a novel quaternionic sampling theorem and formulas, connecting quaternion Fourier analysis with prolate spheroidal wave functions for bandlimited signals.

Findings

Derived new quaternionic sampling formulas.

Linked quaternionic signals with prolate spheroidal wave functions.

Analyzed energy concentration in quaternion Fourier domain.

Abstract

The Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem provides a reconstruction formula for the bandlimited signals. In this paper, a novel kind of the WSK sampling theorem is established by using the theory of quaternion reproducing kernel Hilbert spaces. This generalization is employed to obtain the novel sampling formulas for the bandlimited quaternion-valued signals. A special case of our result is to show that the 2D generalized prolate spheroidal wave signals obtained by Slepian can be used to achieve a sampling series of cube-bandlimited signals. The solutions of energy concentration problems in quaternion Fourier transform are also investigated.