# Bessel interpolation formula

A formula which is defined as half the sum of the Gauss formula (cf. Gauss interpolation formula) for forward interpolation on the nodes at the point : (1) and the Gauss formula of the same order for backward interpolation with respect to the node , i.e. with respect to the population of nodes  (2) Putting Bessel's interpolation formula assumes the form (, ): (3)   Bessel's interpolation formula has certain advantages over Gauss' formulas (1), (2); in particular, if the interpolation is at the middle of the segment, i.e. at , all coefficients at the differences of odd orders vanish. If the last term on the right-hand side of (3) is omitted, the polynomial , which is not a proper interpolation polynomial (it coincides with only in the nodes ), represents a better estimate of the residual term (cf. Interpolation formula) than the interpolation polynomial of the same degree. Thus, for instance, if , the estimate of the last term using the polynomial which is most frequently employed written with respect to the nodes , is almost 8 times better than that of the interpolation polynomial written with respect to the nodes or ().