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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\label{eq-integral-B2}\tag{24.3.1} \end{equation}
\begin{equation} S_3(x) = \int_{x-1/2}^{x+1/2} S_2(t)\,dt\label{eq-integral-B3}\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*}

Quadratic B-splines are awkward to use because of the fractional transition points above, which do not align with grid points. Averaging again by (24.3.2) improves the situation and leads, after tedious case-by-case computations, to the formula (24.2.2).