## Divided Difference

The divided difference on points , , ..., of a function is defined by and

 (1)

for . The first few differences are
 (2) (3) (4)

Defining
 (5)

and taking the Derivative
 (6)

gives the identity
 (7)

Consider the following question: does the property

 (8)

for and a given function guarantee that is a Polynomial of degree ? Aczél (1985) showed that the answer is yes'' for , and Bailey (1992) showed it to be true for with differentiable . Schwaiger (1994) and Andersen (1996) subsequently showed the answer to be yes'' for all with restrictions on or .

See also Newton's Divided Difference Interpolation Formula, Reciprocal Difference

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 877-878, 1972.

Aczél, J. A Mean Value Property of the Derivative of Quadratic Polynomials--Without Mean Values and Derivatives.'' Math. Mag. 58, 42-45, 1985.

Andersen, K. M. A Characterization of Polynomials.'' Math. Mag. 69, 137-142, 1996.

Bailey, D. F. A Mean-Value Property of Cubic Polynomials--Without Mean Values.'' Math. Mag. 65, 123-124, 1992.

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 439-440, 1987.

Schwaiger, J. On a Characterization of Polynomials by Divided Differences.'' Aequationes Math. 48, 317-323, 1994.

© 1996-9 Eric W. Weisstein
1999-05-24