  In this paper, we give a causal solution to the problem of spline
interpolation using H-infinity optimal approximation. Generally speaking,
spline interpolation requires filtering the whole sampled data, the past and
the future, to reconstruct the inter-sample values. This leads to non-causality
of the filter, and this becomes a critical issue for real-time applications.
Our objective here is to derive a causal system which approximates spline
interpolation by H-infinity optimization for the filter. The advantage of
H-infinity optimization is that it can address uncertainty in the input signals
to be interpolated in design, and hence the optimized system has robustness
property against signal uncertainty. We give a closed-form solution to the
H-infinity optimization in the case of the cubic splines. For higher-order
splines, the optimal filter can be effectively solved by a numerical
computation. We also show that the optimal FIR (Finite Impulse Response) filter
can be designed by an LMI (Linear Matrix Inequality), which can also be
effectively solved numerically. A design example is presented to illustrate the
result.
