Re: cubic spline theory problem,

vsgdp wrote:
Let S(x) be the cubic spline interpolating a function f(x), which has
continuous fifth derivative on [a,b], on grid points a = x_0 < x_1 <
... < x_n = b.
Suppose S'(x_0) = f'(x_0) and S'(x_n) = f'(x_n).
Moreover, let g(x) be *any* function with continuous second
derivative such that g(x_j) = f(x_j) for j=0,...,n and that g(x)
satisfies the same end point conditions as S(x).
Show that:
Integral( (S''(x))^2 dx from a to b ) <= Integral( (g''(x))^2 dx from
a to b ).
1. So basically we are showing that the definite integral of the
squared second derivative of the cubic spline S(x) is always less
than or equal to the definite integral of the squared second
derivative of *any* other function?
Intuitively, why is this true?
Is it because the second derivative of S(x) is piecewise linear, and
so a line approximation of the 2nd derivative is the worst?
Why is this statement even useful?
2. Can someone give me a hint on where to begin proving it?

This property is sometimes referred to as the minimum curvature property of
splines and means that the spline approximation has minimal curvature (since
curvature ~ s'') and hence is the smoothest such approximation. The
physical interpretation of the property is that the mechanical spline, a
beam, assumes a shape which minimizes its potential energy. For a beam in
flexure, PE~integral(M^2), where moment M~s''. A mechanical spline is a
flexible strip once used by draftsmen to draw smooth curves. For proofs and
discussion, see, for example, Vandergraft's "Introduction to Numerical
Calculations." BTW, a cubic spline has continuous second derivative, not
fifth derivative.



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