- Title
- Higher-dimensional prolate spheroidal wave functions
- Creator
- Baghal Ghaffari, Hamed
- Relation
- University of Newcastle Research Higher Degree Thesis
- Resource Type
- thesis
- Date
- 2022
- Description
- Research Doctorate - Doctor of Philosophy (PhD)
- Description
- Prolate spheroidal wave functions (PSWFs) are special functions that have long been used in mathematical physics. They are real-valued functions on the line which arise when solving the Helmholtz equation by separation of variables in prolate spheroidal coordinates (playing the role of Legendre polynomials in spherical coordinates). In a beautiful series of papers published in the Bell Labs Technical Journal in the 1960s, [60] Slepian and Pollak observed that these same functions were the solutions to the spectral concentration problem which is of enormous importance in communications technologies. This observation allowed for the efficient computation of PSWFs and the eventual incorporation of their digital counterparts (the discrete PSWFs) in computer hardware. Analyticity and asymptotic properties of PSWFs, including numerical evaluations and applications to quadrature and interpolation are investigated in [7, 13, 34, 40, 48, 53, 56, 65]. Other applications of PSWFs can be found in [20, 30, 35, 38, 41, 42, 47, 57, 64]. In higher dimensions, the computation of PSWFs is more problematic, and the differential equation from which they arise is singular, causing instabilities [59] [45]. Furthermore, the higher dimensional PSWFs, like the one-dimensional PWSFs, are real-valued. Here we seek natural multi-channel versions of the PSWFs of [59] with a view to applications in treating multi-channel signals such as colour images and electromagnetic fields. Clifford analysis is a means through which many of the fundamental theorems and techniques of complex analysis can be lifted to higher dimensions. It is a branch of mathematics that has recently found many applications in signal and image processing. In this thesis, we first review the construction of PSWFs. We also provide some essential features of the PSWFs such as being eigenfunctions of the finite Fourier transformation. The PSWFs are also the solutions to the spectral concentration problem. Considering the importance of PSWFs, we construct multidimensional multichannel PSWFs and investigate their properties. To do this, first, we review some necessary tools in Clifford analysis. We develop further properties of Clifford-Legendre polynomials with particular emphasis on the Bonnet recurrence formula. Using this, we construct Clifford prolate spheroidal wave functions (CPSWFs). We show that the CPSWFs enjoy analogs of the many remarkable features of the one-dimensional PSWFs, including being eigenfunctions of the finite Fourier transformation and solutions of the spectral concentration problem, as well as a spectrum accumulation property. Finally, for the sake of the computation of 3−dimensional CPSWFs, we construct spherical monogenics in R^3. We use two different methods for this construction. In the first method, we apply the Dirac operator to an orthonormal basis of spherical harmonics. The second method employs the reproducing kernel for spherical monogenics of fixed degree and optimization methods. The work in chapters 2 and 3 is based on the papers [5], [3], and [4]. The results of chapter 4 will soon be prepared for publication.
- Subject
- spheroidal wave functions; mathematical physics; Clifford analysis; optimization; PSWF
- Identifier
- http://hdl.handle.net/1959.13/1465850
- Identifier
- uon:47393
- Rights
- Copyright 2022 Hamed Baghal Ghaffari
- Language
- eng
- Full Text
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| Thumbnail | File | Description | Size | Format | |||
|---|---|---|---|---|---|---|---|
| View Details Download | ATTACHMENT01 | Thesis | 7 MB | Adobe Acrobat PDF | View Details Download | ||
| View Details Download | ATTACHMENT02 | Abstract | 282 KB | Adobe Acrobat PDF | View Details Download |