Zernike Polynomials - Higher Dimensions

Higher Dimensions

The concept translates to higher dimensions if multinomials in Cartesian coordinates are converted to hyperspherical coordinates, multiplied by a product of Jacobi Polynomials of the angular variables. In dimensions, the angular variables are Spherical harmonics, for example. Linear combinations of the powers define an orthogonal basis satisfying

.

(Note that a factor is absorbed in the definition of here, whereas in the normalization is chosen slightly differently. This is largely a matter of taste, depending on whether one wishes to maintain an integer set of coefficients or prefers tighter formulas if the orthogonalization is involved.) The explicit representation is

R_n^{(l)}(\rho) = \sqrt{2n+D}\sum_{s=0}^{(n-l)/2} (-1)^s{(n-l)/2 \choose s}{n-s-1+D/2 \choose (n-l)/2}\rho^{n-2s}
=(-1)^{(n-l)/2}\sqrt{2n+D}\sum_{s=0}^{(n-l)/2} (-1)^s{(n-l)/2 \choose s} {s-1+(n+l+D)/2 \choose (n-l)/2} \rho^{2s+l}
=(-1)^{(n-l)/2}\sqrt{2n+D}{ (D+n+l)/2-1 \choose (n-l)/2}\rho^l {}_2F_1( -(n-l)/2,(n+l+D)/2;l+D/2;\rho^2)

for even, else identical to zero.

Read more about this topic:  Zernike Polynomials

Famous quotes containing the words higher and/or dimensions:

    “Le Corbusier was the sort of relentlessly rational intellectual that only France loves wholeheartedly, the logician who flies higher and higher in ever-decreasing circles until, with one last, utterly inevitable induction, he disappears up his own fundamental aperture and emerges in the fourth dimension as a needle-thin umber bird.”
    —Tom Wolfe (b. 1931)

    “It seems to me that we do not know nearly enough about ourselves; that we do not often enough wonder if our lives, or some events and times in our lives, may not be analogues or metaphors or echoes of evolvements and happenings going on in other people?—or animals?—even forests or oceans or rocks?—in this world of ours or, even, in worlds or dimensions elsewhere.”
    —Doris Lessing (b. 1919)