If a wave aberration is expressed by the Zernike polynomial, *R*_{n}^{m}(*r*)cos *m*θ, the aberration average over the pupil is zero. By using the Maréchal balancing technique, however, we always are able to satisfy the above condition: for a rotationally symmetric aberration (*m* = 0), the constant term, which represents a shift in the radius of the reference sphere, should be treated as a variable aberration coefficient. This immediately follows from general considerations given in Sections 3 and 5. See also Ref. 24.

When the truncation ratio is *r*_{0} 1/*n*, the pupil radius is equal to *n* times the 1/*e*^{2} beam radius at the lens.

D. D. Lowenthal, Ref. 12, p. 2132, Fig. 6.

H. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Chap. 9, Eqs. (16) and (17).

Y. L. Luke, Mathematical Functions and Their Approximations (Academic, New York, 1975), Chap. 4, especially Secs. 4.2 and 4.5.

D. D. Lowenthal, Ref. 12, p. 2129.

D. D. Lowenthal, Ref. 12, p. 2132, Eq. (13); note that certain factors on p. 2132 are misprinted.

V. N. Mahajan, Ref. 13, p. 82, especially Eqs. (62), (68), and (75)–(77); "Zernike annular polynomials for imaging systems with annular pupils: errata," J. Opt. Soc. Am. 71, 1408 (1981).

See, for example, D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, New York, 1973), Chaps. 2 and 3.

P. Jacquinot and B. Roizen-Dossier, "Apodisation," in Progress in Optics (North-Holland, Amsterdam, 1964), Vol. III, pp. 31–188.

H. Bateman and A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. II, Chap. 10.

T. S. Chihara, "An Introduction to Orthogonal Polynomials," in Mathematics and Its Applications, A Series of Monographs and Texts (Gordon and Breach, New York, 1978), Vol. 13, especially Chap. 1.

Also note that the Gram determinant [Eq. (4.6)] can be treated as a discriminant of symmetrical quadratic form (4.8); since the form is positive-definite, all the determinants *G*_{p} are positive.

Note that, having the values of appropriate moments, another recursive method of orthogonalization may be also proposed. Namely, any three connective orthogonal polynomials are connected by a single recurrence relation (see Ref. 20, pp. 18 and 19, for example).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal, London, 1966), Chaps. I, V.

A. Maréchal and M. Françon, Diffraction-Structure des Images (Editions de la Revue d'Optique Théorique et Instrumentale, Paris, 1960), Chap. 8.

E. L. O Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963), Chap. 4.