# 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]).

#### References

[1] | I.S. Berezin, N.P. Zhidkov, "Computing methods" , 1 , Pergamon (1973) (Translated from Russian) |

[2] | N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) |

#### Comments

#### References

[a1] | M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1970) |

[a2] | F.B. Hildebrand, "Introduction to numerical analysis" , Addison-Wesley (1956) |

**How to Cite This Entry:**

Bessel interpolation formula. M.K. Samarin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Bessel_interpolation_formula&oldid=12755