Sharp bound on the radial derivatives of the Zernike circle polynomials (disk polynomials)
aa r X i v : . [ m a t h . C A ] O c t Sharp bound on the radial derivatives of theZernike circle polynomials (disk polynomials)
A.J.E.M. JanssenEindhoven University of Technology,Department of Mathematics and Computer Science,P.O. Box 513, 5600 MB Eindhoven, The Netherlands.E-mail address: { [email protected] } Tel.: +31-402474541
Abstract.
We sharpen the bound n k on the maximum modulus of the k th radialderivative of the Zernike circle polynomials (disk polynomials) of degree n to n ( n − ) · ... · ( n − ( k − ) / k (1 / k . This bound is obtained from a resultof Koornwinder on the non-negativity of connection coefficients of the radialparts of the circle polynomials when expanded into a series of Chebyshevpolynomials of the first kind. The new bound is shown to be sharp for, forinstance, Zernike circle polynomials of degree n and azimuthal order m when m = O ( √ n ) by using an explicit expression for the connection coefficients interms of squares of Jacobi polynomials evaluated at 0. Keywords : Zernike circle polynomial, disk polynomial, radial derivative,Chebyshev polynomial, connection coefficient, Gegenbauer polynomial.1
Introduction
The Zernike circle polynomials (ZCPs) Z mn are defined for integer n and m such that n − | m | is even and non-negative by Z mn ( r e iϕ ) = e imϕ R | m | n ( r ) = e imϕ r m P (0 , | m | ) n −| m | (2 r − , (1)where 0 ≤ r ≤ ϕ ∈ R , and where R | m | n ( r ) is its radial part with P ( α,β ) p ( x )the Jacobi polynomial of degree p = 0 , , ... corresponding to the weightfunction (1 − x ) α (1 + x ) β , − ≤ x ≤
1. The Zernike circle polynomials aredirectly related to the disk polynomials R ( α ) k,l , see [8], 18.37(i) on pp. 477–478,by Z mn = R (0) ( n + m ) , ( n − m ) .The ZCPs were introduced by Zernike [10] in connection with his phasecontrast method and further elaborated by Nijboer in his thesis [7] on thediffraction theory of aberrations for optical systems with circular pupils. TheZCPs were also used by Cormack [2] in the context of computerized tomogra-phy with functions on the disk to be reconstructed from their line integrals.The usefulness of the ZCPs in these fields arises from the fact that they are acomplete orthogonal set of functions on the unit disk that have a closed-formexpression for both their 2-D Fourier transform and their Radon transform.For the basic properties of ZCPs and their application in optical diffractiontheory, see [1], Appendix VII and Chapter 9, Section 2. For further mathe-matical properties of the ZCPs, among which an addition-type formula, see[3]. In a recent statistical study of the Radon transform in a medical imag-ing context, there is interest in upper bounds for the modulus of the radialderivatives of the ZCPs, see [6], Appendix B, where it is shown that for k = 0 , , ... max ≤ r ≤ | ( R | m | n ) ( k ) ( r ) | ≤ n k , ≤ r ≤ , (2)with f ( k ) ( r ) denoting the k th derivative of f at r . We shall sharpen thisbound tomax ≤ r ≤ | ( R | m | n ) ( k ) ( r ) | ≤ n ( n − ) · ... · ( n − ( k − )2 k (1 / k , ≤ r ≤ , (3)where ( α ) = 1, ( α ) k = α ( α + 1) · ... · ( α + k − k = 1 , , ... (Pochhammersymbol), and where the right-hand side of (3) is to be interpreted as 1 for thecase that k = 0. Observe that the right-hand side of (3) vanishes when k > n .The inequality | R | m | n ( r ) | ≤
1, 0 ≤ r ≤
1, i.e., the case k = 0 in (3), followsfrom non-negativity of the connection coefficients that occur when R | m | n ( r ) is2xpanded in a series of Chebyshev polynomials of the first kind, as was shownby Koornwinder [5], and the fact that R | m | n (1) = 1. The general case k =1 , , ... follows by explicit results for derivatives of Chebyshev polynomialsin terms of Gegenbauer polynomials, having an explicit expression for themaximum modulus of these. The inequality in (3) is sharp, in the sense thatfor large n and m = O ( √ n ), the inequality is obtained within a factor ofthe order 1 / √ n . This is shown by using a result of Janssen [3], Section 5,that gives the connection coefficients when expanding R | m | n into a series ofChebyshev polynomials in an explicit form. In [3], Section 5, it is shown that R | m | n (cos ϑ ) = ⌊ n/ ⌋ X j =0 a n − j cos( n − j ) ϑ , ≤ ϑ ≤ π , (4)where ⌊ n/ ⌋ is the largest integer not exceeding n/
2, and, for non-negativeinteger i with n − i non-negative and even, a i = a i ( m ) = ε i p ! q ! s ! t ! ( ) l ( P ( γ,ε ) p (0)) , (5)in which ε = 1, ε = ε = · · · = 2, and p = n − l , q = n + l , s = n − r , t = n + r , γ = l − r , δ = l + r , (6)with l = max( | m | , i ) , r = min( | m | , i ) . (7)Note that p , q , s , t , γ , δ in (6) are non-negative and integer since n , m , i have same parity. The a i are non-negative, and this confirms Koornwinder’sresult. Since R | m | n (1) = P (0 , | m | ) ( n −| m | ) (1) = 1, we have | R | m | n (cos ϑ ) | ≤ ⌊ n/ ⌋ X j =0 a n − j = R | m | n (1) = 1 , (8)showing (3) for k = 0.The result (4–7) gives specific information as to when a i = 0. For in-stance, when i = 0, we have l = | m | , r = 0, and so γ = δ , p = ( n − | m | ).3herefore a = 0 ⇔ ( n − | m | ) is odd, since P ( γ,γ ) p (0) = 0 ⇔ p is odd, see [8],second line of Table 18.61 on p. 444. Similarly, when m = 0 we have a i = 0if and only if ( n − i ) is odd.An important case of a non-vanishing a n − j is a n = a n ( m ) (so j = 0) thatis used in Section 3 where we discuss sharpness of the bound in (3). Then l = i = n , r = | m | , p = 0, and since P ( γ,δ )0 (0) = 1 , we thus get a n = 2( ) n (cid:16) nn − | m | (cid:17) , n > . (9)In a similar manner, see [9], bottom of p. 161 for P ( γ,δ )1 (0) and P ( γ,δ )2 (0), a n − = a n | m | n , n − > , n − ≥ | m | ; a n − = a n ( n − | m | ) n ( n − , n − > , n − ≥ | m | . (10) Let k = 1 , , ... . Setting r = cos ϑ in (4) and using that cos( n − j ) ϑ = T n − j (cos ϑ ), with T l the Chebyshev polynomial of the first kind and of degree l = 0 , , .. , we have ( R | m | n ) ( k ) ( r ) = ⌊ n/ ⌋ X j =0 a n − j T ( k ) n − j ( r ) . (11)From [8], 18.9.21 on p. 447, 18.7.4 on p. 444 and 18.9.19 on p. 446 (used k − T ( k ) n − j ( r ) = 2 k − ( k − n − j ) C kn − j − k ( r ) , (12)where C ki is the Gegenbauer polynomial of degree i corresponding to theweight function (1 − x ) k − / , − ≤ x ≤ C ki = 0 for i = − , − , ... .The result (12) can be found in this form in [9], first item in line 6 of p. 188,from which we borrow the notation C ki , rather than the notation C ( k ) i asused in [8], Chapter 18, to avoid confusion with the operation of taking the k th derivative. From [8], 18.14.4 on p. 450, we have that | C ki ( r ) | is maximal(strictly when i >
0) at r = ±
1, with maximum value C ki (1) = (2 k ) i i ! . (13)4ince a n − j ≥
0, we have from (11) and (12) that | ( R | m | n ) ( k ) ( r ) | is maximal at r = 1, with maximal value( R | m | n ) ( k ) (1) = ⌊ n/ ⌋ X j =0 a n − j T ( k ) n − j (1)= 2 k − ( k − ⌊ n/ ⌋ X j =0 a n − j ( n − j ) (2 k ) n − j − k ( n − j − k )! . (14)The maximum at r = 1 is strict when n > a n > i (2 k ) i − k / ( i − k )! is positive and increasing in i = k, k + 1 , ... ,while P ⌊ n/ ⌋ j =0 a n − j = 1 and a n − j ≥ j = 0 , , ..., ⌊ n/ ⌋ . Therefore2 k − ( k − n (2 k ) n − k ( n − k )! a n ≤ ( R | m | n ) ( k ) (1) ≤ k − ( k − n (2 k ) n − k ( n − k )! . (15)The third member in (15) can be rewritten as2 k − ( k − n (2 k ) n − k ( n − k )! = 2 k − ( k − k − n ( n + k − n − k )!= n ( n − ) · ... · ( n − ( k − )2 k (1 / k , (16)and this gives the upper bound (3) as well as a lower bound formax ≤ r ≤ | ( R | m | n ) ( k ) ( r ) | . We have ( R | m | n ) ( k ) (1) = ⌊ n/ ⌋ X j =0 a n − j ( m ) T ( k ) n − j (1) , (17)where T ( k )1 (1) are given for i = k, k + 1 , ... by T ( k ) i (1) = i ( i − · ... · ( i − ( k − )2 k (1 / k = 2 k − ( k − i (2 k ) i − k ( i − k )! , (18)and vanish for i < k . When (for a given n and m ) k increases, the terms a n − j ( m ) in the series in (7) with small j get relatively more weight than those5ith large j . It is therefore expected that the inquality in (3), i.e., the upperbound in (15), tends to be sharp for small k while the lower bound in (15)tends to be sharp for large k . Indeed, there is equality in the second inqualityin (15) when k = 0 and in the first inequality in (15) when k = n, n − j = 0).For the cases k = 1 , R | m | n ) ′ (1) = | m | + 2 w , ( R | m | n ) ′′ (1) = | m | ( | m | −
1) + 2 w ( w + | m | − , (19)where w = ( n −| m | )( n + | m | +2). With n , n ( n −
1) being the right-handside of (3) for k = 1 , R | m | n ) ′ (1) n → , ( R | m | n ) ′′ (1) n ( n − →
38 (20)when n → ∞ and m = o ( n ). The identities in (19) follow from( R | m | n ) ′ ( r ) = ( n − −| m − | ) X j =0 ( n − j ) R | m − | n − − j ( r )+ ( n − −| m +1 | ) X j =0 ( n − j ) R | m +1 | n − − j ( r ) , (21)see [4], (10) and add the ± -cases that occur there, where we also recall that R | m ± | n − − j (1) = 1 for j in the summation ranges in the two series in (21).Actually, (21) was used in [6], Appendix B, to show the bound in (2) byinduction. In general, it can be shown by induction from (21) that the ratioof ( R | m | n ) ( k ) (1) and the right-hand side of (3) converges to (1 / k / (1) k as n → ∞ and m , k are fixed.In the case that k = n −
2, we have( R | m | n ) ( n − (1) = a n ( m ) T ( n − n (1) + a n − ( m ) T ( n − n − (1)= a n ( m ) T ( n − n (1) (cid:16) | m | n n (2 n − (cid:17) , (22)where the first item in (10) and the second form in (18) for T ( n − n − (1) and T ( n − n (1) has been used. Hence, the lower bound in (15) is also sharp for k = n − n → ∞ , and a similar thing can be concluded for k = n − a n ( m ) as given by (9) for n >