Keys Function

The principal reason for the popularity enjoyed by the family of Keys' functions  is the fact that they perform better than linear interpolation, while being interpolating. Thus, they do not require the determination of interpolation coefficients, and the classical Eq. (1) can be used. These functions are made of FIGURE 13 Keys' interpolant with a = -p

FIGURE 12 o-Moms of third degree (central part).

FIGURE 13 Keys' interpolant with a = -p piecewise cubic polynomials and depend on a parameter a (Fig. 13). Their general expression is

{(a + 2)|x|3 -(a + 3)|x |2 + 1 0 <|x| < 1 a\x\3 - 5a|x|2 + 8a|x|-4a 1 <|x|<2 0 2 < |x|.

Comparing this expression to that of the cubic spline, it is apparent that both require the computation of piecewise polynomials of the same support. However, their approximation orders differ: The best order that can be reached by a Keys' function is 3, for the special value a = —s while the cubic spline has order 4. This extra order for ft3 comes at the cost of the computation of a sequence {ck}, for use in Eq. (4). However, by using a recursive filtering approach, this cost can be made negligible. Altogether, the gain in speed offered by Keys' function is not enough to counterbalance the loss in quality when comparing ft3 and tfci. Moreover, the regularity of Keys is C1, which is one less than that of the cubic spline.