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Piecewise Interpolation: Zero Degree Spline Interpolation

0 Degree Spline Interpolation

Given a set of k data points (x_1,y_1),(x_2,y_2),\cdots, (x_k,y_k), a spline of degree 0 can be defined as:

    \[ y=\begin{cases} y_1,&x\in [x_1, x_2) \\ y_2,&x\in [x_2, x_3) \\ \vdots & \vdots \\ y_{k-1},&x\in [x_{k-1}, x_k) \\ y_k, & x=x_k \end{cases} \]

In the previous spline definition, the interpolated value in the interval [x_{i-1},x_i) is equal to the data value y_{i-1}. Alternatively, the interpolated value in the interval (x_{i-1},x_i] can be set equal to the data value y_i. In this case, y can be defined as follows:

    \[ y=\begin{cases} y_1,&x=x_1\\ y_2,&x\in (x_1, x_2] \\ \vdots \vdots \\ y_k,&x\in (x_{k-1}, x_k] \end{cases} \]

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