Section 24.3 Construction of cardinal B-splines
The formula (24.2.2) deserves some explanation. The idea is to start with the linear interpolant (degree 1 spline) \(S_1=L\) defined by (24.2.1) and repeatedly smoothen it using a moving average with a window of size \(1\text{:}\)
\begin{equation}
S_2(x) = \int_{x-1/2}^{x+1/2} S_1(t)\,dt\tag{24.3.1}
\end{equation}
\begin{equation}
S_3(x) = \int_{x-1/2}^{x+1/2} S_2(t)\,dt\tag{24.3.2}
\end{equation}
and so on. The fundamental theorem of calculus shows that
\begin{equation*}
S_2'(x) = S_1(x+1/2)-S_1(x-1/2)
\end{equation*}
in particular \(S_2\) has continuous first derivative. Similarly,
\begin{equation*}
S_3'(x) = S_2(x+1/2)-S_2(x-1/2)
\end{equation*}
so
\begin{equation*}
S_3''(x) = S_2'(x+1/2)-S_2'(x-1/2)
\end{equation*}
is continuous as well.
The function \(S_3\) defined by (24.3.2) is the approximating cubic spline discussed above, while \(S_2\) is a quadratic spline. But these integral formulas should be simplified for practical use. By linearity of integrals,
\begin{equation*}
S_2(x) = \sum_{k=1}^n y_k \int_{x-1/2}^{x+1/2} H(t)\,dt
\end{equation*}
where the integral can be evaluated explicitly: it is a cardinal B-spline of degree 2, which can be written out
\begin{equation*}
\int_{x-1/2}^{x+1/2} H(t)\,dt = \begin{cases}
0 & x \le -3/2 \\
(x+3/2)^2 /2 & -3/2 \le x \le -1/2 \\
3/4 - x^2 & -1/2 \le x \le 1/2 \\
(x-3/2)^2 /2 & 1/2 \le x \le 3/2 \\
0 & x \ge 3/2
\end{cases}
\end{equation*}
