Zou, Cuiming; Kou, Kit Ian; Morais, Joao Prolate spheroidal wave functions associated with the quaternionic Fourier transform. (English) Zbl 1395.42019 Math. Methods Appl. Sci. 41, No. 11, 4003-4020 (2018). Summary: 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 quaternionic signal whose time-frequency energy distribution is most concentrated in a given time-frequency domain. The present paper finds a new kind of quaternionic signals whose energy concentration is maximal in both time and frequency under the quaternionic Fourier transform. The new signals are a generalization of the classical prolate spheroidal wave functions to a quaternionic space, which are called the quaternionic prolate spheroidal wave functions. The purpose of this paper is to present the definition and fundamental properties of the quaternionic prolate spheroidal wave functions and to show that they can reach the extreme case within the energy concentration problem both from the theoretical and experimental description. The superiority of the proposed results can be widely applied to the application of 4D valued problems. In particular, these functions are shown as an effective method for bandlimited quaternionic signals relying on the extrapolation problem. Cited in 2 Documents MSC: 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 94A12 Signal theory (characterization, reconstruction, filtering, etc.) Keywords:quaternionic analysis; quaternionic Fourier transform; quaternionic signal; the energy concentration roblem; Mathieu functions; bandlimited extrapolation PDFBibTeX XMLCite \textit{C. Zou} et al., Math. Methods Appl. Sci. 41, No. 11, 4003--4020 (2018; Zbl 1395.42019) Full Text: arXiv