A relation between Zernike polynomials and other orthogonal polynomials: A review

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Al Tamimi K.J.K.

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Journal Paper

Paper Information

Journal: Caspian Journal of Mathematical Sciences Year:2024 | Volume:13 | Issue:2 Page(s): 386-400

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Title

A relation between Zernike polynomials and other orthogonal polynomials: A review

Author(s)

Al Tamimi K.J.K. | Issue Writer Certificate

Keywords

Zernike polynomials

‎O‎rthogonal polynomials

Legendre polynomials

Jaccobi polynomials

Bessel functions

Abstract

Zernike circle polynomials use for wavefront analysis because of their orthogonality over a circular. ‏‎ So far, different representations have been presented for these polynomials. In this paper, ‎ ‎‎we introduce Zernike polynomials and their important properties. Also, we present some other orthogonal polynomials such ‎as‎ Bessel ‎functions, ‎ Legendre polynomials and ‎ Jaccobi polynomials ‎and ‎give some important results of them‏. ‎‎ Finally, we express the relationship between Zernike orthogonal polynomials and these well-known orthogonal polynomials. ‏‎ For different cases, the ‎radial‎ polynomials in the Zernike functions are obtained using the well-known orthogonal polynomials‏. These relations frequently used for representing‎ wavefront ‎aberrations.

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