Interpolation formula of Gauss.

Interpolation formula of Gauss.

where is a trigonometric polynomial of degree such that for ,..., , and

Interpolation formula of Stirling.

Stirling’s interpolation formula looks like:

where, as before, .

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 ([1], [2]):

(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 ([2]).

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