On Sampling a Function and its Derivatives

Whittaker 's sampling theorem (Whittaker, 1915; McNamee et al., 1971) for the reconstruction of a frequency-band-limited time-function g(t), f rom its uniform (i.e., equidistant) samples g(k); k ~ J (the class of integers); has been extended by Fogel (1955) and Jagerman and Fogel (1956) to include the simultaneous sampling of g(t) and its first derivative. Fogel's work (1955) is quite incomplete in its attempt to treat higher derivatives. Jagerman and Fogel (1956) give an explicit solution for uniform sampling of a function and its first derivative and also extend the results to the complex domain by a method which, in principle, could be used for higher derivatives but cannot readily be adapted to more complicated situations. Kohlenberg (1953) and Yen (1956) consider periodic sampling of g(t). Yen (1956) also discusses arbitrary shifts of a finite number of sample points from their periodic locations. We treat here, more generally, the reconstruction ofg(t) f rom gm(k + h~); l : 0,..., n 1 ;0 ~<t~ < 1 ;p = 1,..., m~ ; h e J. The resulting expression is of the form