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Constructing prolate spheroidal quaternion wave functions on the sphere
Morais J.1; Kou K.I.2
Source PublicationMathematical Methods in the Applied Sciences
ISSN10991476 01704214

Over the last years, considerable attention has been paid to the role of the prolate spheroidal wave functions (PSWFs) introduced in the early sixties by D. Slepian and H.O. Pollak to many practical signal and image processing problems. The PSWFs and their applications to wave phenomena modeling, fluid dynamics, and filter design played a key role in this development. In this paper, we introduce the prolate spheroidal quaternion wave functions (PSQWFs), which refine and extend the PSWFs. The PSQWFs are ideally suited to study certain questions regarding the relationship between quaternionic functions and their Fourier transforms. We show that the PSQWFs are orthogonal and complete over two different intervals: the space of square integrable functions over a finite interval and the three-dimensional Paley–Wiener space of bandlimited functions. No other system of classical generalized orthogonal functions is known to possess this unique property. We illustrate how to apply the PSQWFs for the quaternionic Fourier transform to analyze Slepian's energy concentration problem. We address all of the aforementioned and explore some basic facts of the arising quaternionic function theory. We conclude the paper by computing the PSQWFs restricted in frequency to the unit sphere. The representation of these functions in terms of generalized spherical harmonics is explicitly given, from which several fundamental properties can be derived. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. Copyright © 2016 John Wiley & Sons, Ltd.

Keyword30c65 Prolate Spheroidal Wave Functions Quaternionic Analysis Quaternionic Fourier Transform Quaternionic Functions Spherical Harmonics Subclass 30g35 The Energy Concentration Problem
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Indexed BySCIE
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000384079700004
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Cited Times [WOS]:3   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
Affiliation1.Instituto Tecnológico Autonómo de México
2.Universidade de Macau
Recommended Citation
GB/T 7714
Morais J.,Kou K.I.. Constructing prolate spheroidal quaternion wave functions on the sphere[J]. Mathematical Methods in the Applied Sciences,2016,39(14):3961-3978.
APA Morais J.,&Kou K.I..(2016).Constructing prolate spheroidal quaternion wave functions on the sphere.Mathematical Methods in the Applied Sciences,39(14),3961-3978.
MLA Morais J.,et al."Constructing prolate spheroidal quaternion wave functions on the sphere".Mathematical Methods in the Applied Sciences 39.14(2016):3961-3978.
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