Causal cardinal spline window

For $P=1$ set $W_1(u)={\mathbf{1}}_{[0,1]}(u)$, including both endpoints. For an integer $P\ge 2$, define the causal order-$P$ window by $W_P(u)=\{\begin{array}{ll}{\displaystyle \frac{1}{(P-1)!}\sum _{j=0}^P(-1{)}^j\left(\genfrac{}{}{0px}{}{P}{j}\right)(u-j{)}_{+}^{P-1},}& 0\le u\le P,\\ 0,& u<0\text{ or }u>P\mathrm{.}\end{array}$ Here $s_{+}=\max (s,0)$, $m!$ is the factorial of the nonnegative integer $m$ with $0!=1$, and for integers $0\le j\le P$, $\left(\genfrac{}{}{0px}{}{P}{j}\right)=P!\mathrm{/}(j!(P-j)!)$. Convolution means the integral operation $(f\ast g)(u)={\int }_{\mathbb{R}}f(u-s)g(s)ds,$ when the integrand is measurable and integrable, as in Convolution of two functions on ${\mathbb{R}}^n$. The displayed formula is finite at every $u$ and directly defines $W_P$ to vanish outside $[0,P]$; the local alias lemma proves that it agrees with the $P$-fold convolution of $W_1$ and establishes the resulting integral, positivity, partition, Fourier-transform, and derivative identities.