# Scattered data interpolation by box splines

## Zuowei Shen and Shayne Waldron

## Abstract:

Given scattered data in $\Rs$, interpolation from a dilated box spline space
$S_M(2^k\cdot)$ is always possible for a fine enough scaling.
For example, for the Lagrange function of a point $\gth$
one could take any shifted dilate $M(2^k\cdot-j)$ which is nonzero
at $\gth$ and zero at the other interpolation points.
However, the resulting interpolant, though smooth (and local), will consist of
a set of ``bumps'', and so by any reasonable measure provides a poor
representation of the shape of the underlying function.
On the other hand, it is possible to choose a space of interpolants which
contains some $M(2^k\cdot-j)$ of arbitrarily large support.
But the resulting methods are increasingly less local, and in general
still require some splines with a much higher level of dilation.
Here we provide a multilevel method which constructs a space of
interpolants by taking as many splines as possible from a given dilation
level, then as many from the next (higher) dilation level, and so forth.
The choice at each level is made using the suggestion of \cite{W99},
which is based on the Riesz representation theorem.
This requires an inner product on the ground space $S_M$, and
the higher levels $S_M(2^k\cdot)\ominus S_M(2^{k-1}\cdot)$, $k=1,2,\ldots$.
The inner products used here involve the box spline coefficients, and
prewavelet coefficients of \cite{RS92},
respectively, and are norm equivalent to
$\norm{\cdot}_{L_2(\Rs)}$.
These lead to a scheme which is easily implemented, and numerically stable.
Previously, box spline interpolants have been considered only for data on a
regular grid.

**Keywords:**
box spline,
multilevel approximation,
adaptive methods

**Math Review Classification:**
Primary 65D05, 65D07;
Secondary 42C40

**Length:** 23 pages

**Last Updated:** 10 October 2007

## Availability: