Error estimation for spline interpolation

Can you please indicate a reference for the proof of the fact that the error when interpolating a $C^4$ function by a cubic spline is bounded by $Ch^4\sup_{[x_{i},x_{i+1}]} |f''''(x)|$?

Solution 1:

I'm 90% sure you can find it in deBoor's book "A Practical Guide to Splines".

Here's a link to Amazon.

I checked, and equation (12) on page 55 of the 2001 edition says that: $$ \left\Vert g - I_4g \right\Vert \le \frac{1}{16}\left\vert \tau \right\vert^4 \left\Vert g^{(4)} \right\Vert $$

Here $I_4g$ is the complete cubic spline interpolant of $g$ on the mesh $\tau$.

The result was improved by Hall and Meyer, Journal of Approximation Theory, Vol. 16, 1976, pp.105-122. They showed that the constant $\frac{1}{16}$ could be replaced by $\frac{5}{384}$, and that this constant is best possible.

Solution 2:

In Douglas Arnold's notes: A Concise Introduction to Numerical Analysis.

A proof of error bound is given for Cubic spline from Page 31 exploiting the commutative diagram, the notation $M^k_s(\mathcal{T})$ just means degree $k$ piecewise polynomial and $s$-continuously differentiable.