%for a more readable version of this page.
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Since
$$%
\dvd{t_0,\ldots,t_k}f = \int M(\cdot|t_0,\ldots,t_k) D^kf/k!,
$$%
with $M$ the B-spline normalized to have integral 1, some integrals of the form
$$%
\int M(\cdot|t_0,\ldots,t_k) g
$$%
can (and certainly have been) computed from knowledge of divided differences,
as long as it is easy to obtain $D^{-k}g$ and compute its divided difference.
%
This is certainly the case for the power function,
$$%
g=()^n:x\mapsto x^n,
$$%
for which
$$%
D^{-k}g = ()^{n+k} n!/(n+k)!
$$%
while (see pagep101) %basic divided difference formulae
$$%
\dvd{t_0,\ldots,t_k}()^{n+k} = \sum (t^\alpha: |\alpha|=n, \alpha \ge 0).
$$%
Therefore, the $n$th moment of the B-spline is
$$%
\int ()^n M(\cdot|t_0,\ldots,t_k) = \dvd{t_0,\ldots,t_k}()^{n+k}/{\scriptstyle{n+k\choose n}} =
\sum (t^\alpha: |\alpha|=n, \alpha \ge 0)/{\scriptstyle{n+k\choose n}}\,.
$$%
%
In the same way, since $D^k \exp(-\ii(\cdot)\xi) =
(-\ii \xi)^k\exp(-\ii(\cdot)\xi)$, the Fourier transform of the B-spline is
$$%
\int e^{-\ii(\cdot)\xi}M(\cdot|t_0,\ldots,t_k) =
(k!/(-\ii \xi)^k)\dvd{t_0,\ldots,t_k}e^{-\ii(\cdot)\xi}.
$$%
An early reference (other than Schoenberg's use of this connection throughout
his papers on B-splines, including the very first one, in '46), is
%
%Neuman81b % larry %\refJ Neuman, E.; Moments and Fourier transforms of B-splines; J.Comput.Applied Math.; 7; 1981; 51--62; % % \bye