  One of the fundamental problems in communications is finding the energy
distribution of signals in time and frequency domains. It should, therefore, be
of great interest to find the most energy concentration hypercomplex signal.
The present paper finds a new kind of hypercomplex signals whose energy
concentration is maximal in both time and frequency under quaternionic Fourier
transform. The new signals are a generalization of the prolate spheroidal wave
functions (also known as Slepian functions) to quaternionic space, which are
called quaternionic prolate spheroidal wave functions. The purpose of this
paper is to present the definition and properties of the quaternionic prolate
spheroidal wave functions and to show that they can reach the extreme case in
energy concentration problem both from the theoretical and experimental
description. In particular, these functions are shown as an effective method
for bandlimited signals extrapolation problem.
