Let x0<x1<⋯<xn. For each j=0, …, n, the j^th cardinal spline Cj(x) is defined to be the natural

Step 1: **Construct the Cardinal Splines**u000ATo construct the cardinal splines $C_j(x)$ for $j u003D 0

Let x0<x1<⋯<xn. For each j=0, …, n, the j^th cardinal...

Let $x_0<x_1<\cdots<x_n$. For each $j=0, \ldots, n$, the $j^{\text {th }}$ cardinal spline $C_j(x)$ is defined to be the natural cubic spline interpolating the Lagrange data $$ y_0=0, \quad y_1=0, \quad \ldots \quad y_{j-1}=0, \quad y_j=1, \quad y_{j+1}=0, \quad \ldots \quad y_n=0 . $$ (a) Construct and graph the natural cardinal splines corresponding to the nodes $x_0=0$, $x_1=1, x_2=2$, and $x_3=3$. (b) Prove that the natural spline that interpolates the data $y_0, \ldots, y_n$ can be uniquely written as a linear combination $u(x)=y_0 C_0(x)+y_1 C_1(x)+$ $\cdots+y_n C_n(x)$ of the cardinal splines. (c) Explain why the space of natural splines on $n+1$ nodes is a vector space of dimension $n+1$. (d) Discuss briefly what modifications are required to adapt this method to periodic and to clamped splines.

Let $x_0<x_1<\cdots<x_n$. For each $j=0, \ldots, n$, the $j^{\text {th }}$ cardinal spline $C_j(x)$ is defined to be the natural cubic spline interpolating the Lagrange data
$$
y_0=0, \quad y_1=0, \quad \ldots \quad y_{j-1}=0, \quad y_j=1, \quad y_{j+1}=0, \quad \ldots \quad y_n=0 .
$$
(a) Construct and graph the natural cardinal splines corresponding to the nodes $x_0=0$, $x_1=1, x_2=2$, and $x_3=3$. (b) Prove that the natural spline that interpolates the data $y_0, \ldots, y_n$ can be uniquely written as a linear combination $u(x)=y_0 C_0(x)+y_1 C_1(x)+$ $\cdots+y_n C_n(x)$ of the cardinal splines. (c) Explain why the space of natural splines on $n+1$ nodes is a vector space of dimension $n+1$. (d) Discuss briefly what modifications are required to adapt this method to periodic and to clamped splines.

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