An optimal recovery approach to interpolation

A filter class is defined as a ball in an inner product space, and some standard results on inner product spaces are applied to filter classes. The filter design problem is addressed. The theory of optimal recovery is reviewed, and the interpolation problem is examined within an optimal recovery context. It is shown that the interpolation problem can be reduced to studying a hypercircle in an inner product space. The notion of the Chebyshev center of a set is introduced, and it is noted that the solution to the interpolation problem, from the optimal recovery viewpoint, is to find the Chebyshev center of the hypercircle. The interpolation filter is then the operator that transforms the vector of known samples into the center of the hypercircle. Some auxiliary results such as the linearity and time invariance of the interpolation filter are deduced. It is then shown that the estimation of an unknown sample is the same as the problem of approximating the representer of the unknown sample by a linear combination of the representers of the known samples. Hence the interpolation problem is equivalent to minimizing the L² norm of the error frequency response, from the filter design point of view