%for a more readable version of this page.
%
The (centered) cardinal B-spline of order $k$ was introduced in
Schoenberg46a(page 68) as the inverse Fourier transform
$$%
M_k(x) := \int_{-\infty}^\infty\left(\sin u/2\over u/2\right)^k
e^{\ii ux}\dd u/(2\pi)
$$%
of a certain function having zeros of order $k$ at all points in
$2\pi\ZZ\bs0$
and shown directly to equal the $k$th order central difference of the
(normalized) truncated power of order $k$, i.e.,
$$%
M_k(x) = \gd^k x_+^{k-1}/(k-1)!
= \sum_{j=0}^k (-1)^j {k\choose j} (x+{k\over2}-j)_+^{k-1} /(k-1)!,
$$%
as well as the convolution product
$$%
M_k = M_1*\cdots*M_1
$$%
of $k$ copies of the characteristic
function of the interval $[-1/2\fromto 1/2]$ (see loc.cit.~page~69).
This latter fact gives at once the positivity of $M_k$ on its support,
$(-k/2\fromto k/2)$, as well as the fact that
$$%
\sum_{j\in\ZZ}M_k(\cdot-j) = 1.
$$%
%
Schoenberg later (e.g., in his sequence of papers on cardinal spline
interpolation, summarized in his book Schoenberg73a) also uses the forward
B-spline
$$%
Q_k(x) = M_k(x+k/2) = M(x|0,\ldots,k)
$$%
and observes, on page 61 of that book, that
$$%
\int_{-\infty}^\infty Q_m(x-j)Q_m(x-k)\dd x = M_{2m}(j-k).
$$%
%
%
\bye