Divided Difference
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover,
الجزء والصفحة:
...
28-11-2021
2713
Divided Difference
The divided difference ![f[x_0,x_1,x_2,...,x_n]](https://mathworld.wolfram.com/images/equations/DividedDifference/Inline1.gif)
, sometimes also denoted ![[x_0,x_1,x_2,...,x_n]](https://mathworld.wolfram.com/images/equations/DividedDifference/Inline2.gif)
(Abramowitz and Stegun 1972), on 
points 
, 
, ..., 
of a function 
is defined by ![f[x_0]=f(x_0)](https://mathworld.wolfram.com/images/equations/DividedDifference/Inline8.gif)
and
![f[x_0,x_1,...,x_n]=(f[x_0,...,x_(n-1)]-f[x_1,...,x_n])/(x_0-x_n)](https://mathworld.wolfram.com/images/equations/DividedDifference/NumberedEquation1.gif) |
(1)
|
for 
. The first few differences are
Defining
 |
(5)
|
and taking the derivative
 |
(6)
|
gives the identity
 |
(7)
|
Consider the following question: does the property
![f[x_1,x_2,...,x_n]=h(x_1+x_2+...+x_n)](https://mathworld.wolfram.com/images/equations/DividedDifference/NumberedEquation5.gif) |
(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 
.
REFERENCES:
Abramowitz, M. and Stegun, I. 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.
Jeffreys, H. and Jeffreys, B. S. "Divided Differences." §9.012 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 260-264, 1988.
Schwaiger, J. "On a Characterization of Polynomials by Divided Differences." Aequationes Math. 48, 317-323, 1994.
Sauer, T. and Xu, Y. "On Multivariate Lagrange Interpolation." Math. Comput. 64, 1147-1170, 1995.
Whittaker, E. T. and Robinson, G. "Divided Differences" and "Theorems on Divided Differences." §11-12 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 20-24, 1967.
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