Bound on the radial derivatives of the Zernike circle polynomials (disk polynomials)
| Autor: | Augustus J. E. M. Janssen |
|---|---|
| Přispěvatelé: | Stochastic Operations Research |
| Rok vydání: | 2020 |
| Předmět: |
Chebyshev polynomials
Degree (graph theory) Series (mathematics) Zernike polynomials Zernike circle polynomial General Mathematics 010102 general mathematics Order (ring theory) Radial derivative 010103 numerical & computational mathematics Connection coefficient Disk polynomial Chebyshev polynomial 01 natural sciences Connection (mathematics) Combinatorics symbols.namesake symbols Gegenbauer polynomial Jacobi polynomials 0101 mathematics Mathematics |
| Zdroj: | Indagationes Mathematicae, 31(5), 842-847. Elsevier |
| ISSN: | 0019-3577 |
| DOI: | 10.1016/j.indag.2020.04.001 |
| Popis: | We sharpen the bound n 2 k on the maximum modulus of the k th radial derivative of the Zernike circle polynomials (disk polynomials) of degree n to n 2 ( n 2 − 1 2 ) ⋅ . . . ⋅ ( 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 attained within a factor O ( 1 ∕ n ) for 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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