A bell-shaped or B-spline u=β(x) interpolates the data β(-2)=0, β(-1)=1, β(0)=4, β(1)=1,

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A bell-shaped or B-spline u=β(x) interpolates the data...

A bell-shaped or $B$-spline $u=\beta(x)$ interpolates the data $$ \beta(-2)=0, \quad \beta(-1)=1, \quad \beta(0)=4, \quad \beta(1)=1, \quad \beta(2)=0 . $$ (a) Find the explicit formula for the natural $B$-spline and plot its graph. (b) Show that $\beta(x)$ also satisfies the homogeneous clamped boundary conditions $u^{\prime}(-2)=u^{\prime}(2)=0$. (c) Show that $\beta(x)$ also satisfies the periodic boundary conditions. Thus, for this particular interpolation problem, the natural, clamped, and periodic splines happen to coincide. (d) Show that $\beta^{\star}(x)=\left\{\begin{array}{ll}\beta(x), & -2 \leq x \leq 2 \\ 0, & \text { otherwise, }\end{array}\right.$ defines a $\mathrm{C}^2$ spline on every interval $[-k, k]$.

A bell-shaped or $B$-spline $u=\beta(x)$ interpolates the data
$$
\beta(-2)=0, \quad \beta(-1)=1, \quad \beta(0)=4, \quad \beta(1)=1, \quad \beta(2)=0 .
$$
(a) Find the explicit formula for the natural $B$-spline and plot its graph. (b) Show that $\beta(x)$ also satisfies the homogeneous clamped boundary conditions $u^{\prime}(-2)=u^{\prime}(2)=0$.
(c) Show that $\beta(x)$ also satisfies the periodic boundary conditions. Thus, for this particular interpolation problem, the natural, clamped, and periodic splines happen to coincide.
(d) Show that $\beta^{\star}(x)=\left\{\begin{array}{ll}\beta(x), & -2 \leq x \leq 2 \\ 0, & \text { otherwise, }\end{array}\right.$ defines a $\mathrm{C}^2$ spline on every interval $[-k, k]$.

Key Concepts

Periodic Splines

Periodic splines enforce conditions such that the function and its derivatives are equal at the endpoints of the interval, thereby producing a smooth, repeating pattern. This approach is useful when the data or problem inherently implies a periodic structure.

C2 Continuity

C2 continuity in the context of splines means that the function, its first derivative, and its second derivative are all continuous across the intervals. This degree of smoothness is crucial for ensuring that transitions between polynomial pieces do not exhibit sudden changes in curvature.

Clamped Boundary Conditions

Clamped boundary conditions require that the first derivative of the spline is specified at the endpoints of the interpolation interval. Homogeneous clamped conditions specifically set these derivatives to zero, ensuring a flat tangent at the boundaries and providing additional smoothness.

B-spline Interpolation

B-spline interpolation involves using basis splines (B-splines) to construct a smooth, piecewise polynomial function that approximates or exactly interpolates given data points. These splines are valued for their local control, compact support, and the ease with which they can be manipulated to obtain smooth curves.

Natural Splines

Natural splines are a type of spline interpolation where the second derivative is set to zero at the boundary points. This boundary condition minimizes the bending energy, yielding a smooth curve that tends to be less affected by oscillations at the boundaries.

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