OMoms

The o-Moms functions are indexed by their polynomial degree n. They are symmetric and their knots are identical to those of the B-spline they descend from. Moreover, they have the same support as ¿6n, that is, W = n + 1; this support is the smallest achievable for a synthesis function with approximation order L = n + 1. Although their order is identical to that of a B-spline of same degree, their approximation error constant Cv is much smaller. In fact, the o-Moms functions are such that their least-squares constant reaches its smallest possible value. In this sense, they are asymptotically optimal approxi-mators, being the shortest for a given support, with the highest approximation order, and the smallest approximation error constant .

These functions are not interpolating; thus, we need a way to compute the sequence of coefficients {q} required for the implementation of Eq. (4). Fortunately, the same routine as for the B-splines can be used (see Appendix).

The o-Moms functions of degree zero and one are identical to ¿6° and 61, respectively. The o-Moms functions of higher degree can be determined recursively in the Fourier domain ; we give here the expression for n = 3: FIGURE 11 o-Moms of third degree. (Left)

1 d2

n |x|3 — |x|2+14 |x|+2f o <|x|<1 = j —f |x|3 + |x |2 — 4| |x|+f 1 < |x| <2 I 0 2 < |x|.

As a curiosity, we point out in Figs 11 and 12 that this synthesis function has a slope discontinuity at the origin; thus, its regularity is C° (in addition, it has other slope discontinuities for |x| = 1 and |x| = 2.) It is nevertheless optimal in the sense described.