Sharp bound on the radial derivatives of the Zernike circle polynomials (disk polynomials)

2019/10/16 by Janssen, A. J. E. M.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1910.07250

Abstract

We sharpen the bound n^2k on the maximum modulus of the k^{\rm th} radial derivative of the Zernike circle polynomials (disk polynomials) of degree n to n²(n²-1²)⋅ ... ⋅(n²-(k-1)²)/2^k(1/2)_k. This bound is obtained from a result of Koornwinder on the non-negativity of connection coefficients of the radial parts of the circle polynomials when expanded into a series of Chebyshev polynomials of the first kind. The new bound is shown to be sharp for, for instance, Zernike circle polynomials of degree n and azimuthal order m when m=O(√(n)) by using an explicit expression for the connection coefficients in terms of squares of Jacobi polynomials evaluated at 0. Keywords: Zernike circle polynomial, disk polynomial, radial derivative, Chebyshev polynomial, connection coefficient, Gegenbauer polynomial.

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