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

TL;DR

This paper derives a sharper, explicit bound on the maximum modulus of the radial derivatives of Zernike circle polynomials, improving previous estimates and demonstrating sharpness in certain cases.

Contribution

It provides a new, precise bound on the radial derivatives of Zernike polynomials using connection coefficients and Chebyshev polynomial expansions, advancing theoretical understanding.

Findings

New bound on radial derivatives of Zernike polynomials

Bound is sharp for certain degrees and azimuthal orders

Utilizes connection coefficients and Chebyshev polynomial expansions

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 $n^2(n^2-1^2)\cdot ... \cdot(n^2-(k-1)^2)/2^k(1/2)_k$. 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 $m=O(\sqrt{n})$ 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.