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Quaternion Zernike spherical polynomials

Authors: J. Morais and I. Cação
Journal: Math. Comp. 84 (2015), 1317-1337
MSC (2010): Primary 26C05, 30G35; Secondary 33C45, 42C05
Published electronically: August 29, 2014
MathSciNet review: 3315510
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Abstract: Over the past few years considerable attention has been given to the role played by the Zernike polynomials (ZPs) in many different fields of geometrical optics, optical engineering, and astronomy. The ZPs and their applications to corneal surface modeling played a key role in this development. These polynomials are a complete set of orthogonal functions over the unit circle and are commonly used to describe balanced aberrations. In the present paper we introduce the Zernike spherical polynomials within quaternionic analysis ((R)QZSPs), which refine and extend the Zernike moments (defined through their polynomial counterparts). In particular, the underlying functions are of three real variables and take on either values in the reduced and full quaternions (identified, respectively, with $ \mathbb{R}^3$ and $ \mathbb{R}^4$). (R)QZSPs are orthonormal in the unit ball. The representation of these functions in terms of spherical monogenics over the unit sphere are explicitly given, from which several recurrence formulae for fast computer implementations can be derived. A summary of their fundamental properties and a further second order homogeneous differential equation are also discussed. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach.

(R)QZSPs are new in literature and have some consequences that are now under investigation.

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Additional Information

J. Morais
Affiliation: Center for Research and Development in Mathematics and Applications, University of Aveiro, 3810-193 Aveiro, Portugal.

I. Cação
Affiliation: Department of Mathematics, Center for Research and Development in Mathematics and Applications, University of Aveiro, 3810-193 Aveiro, Portugal.

Keywords: Quaternionic analysis, Ferrer's associated Legendre functions, Chebyshev polynomials, Zernike polynomials, spherical aberrations, spherical monogenics.
Received by editor(s): March 13, 2013
Received by editor(s) in revised form: August 14, 2013
Published electronically: August 29, 2014
Additional Notes: This work was supported by Portuguese funds through the CIDMA – Center for Research and Development in Mathematics and Applications, and the Portuguese foundation for Science and Technology (“FCT–Fundação para a Ciência e a tecnologia”), within project PEst-OE/MAT/UI4106/2014. Support from FCT via the post-doctoral grant SFRH/BPD/66342/2009 is also acknowledged by the first author
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