AN OPTIMAL RECOVERY APPROACH TO INTERPOLATION

The design of interpolation filters is reviewed. The Chebyshev and integrated squared-error minimization methods are shown to be optimal for certain classes of input signals. We show that filters designed by integrated squared-error minimization in the frequency domain always interpolate the known samples. The interpolation problem is viewed as a problem of recovering the missing samples of a signal. This provides a framework for studying the interpolation problem for more general signal classes, and gives insight into how some of the known results in interpolation filter design arise. For example, it is shown that if the class of input signals is the intersection of a linear variety and a ball in an inner product space, then the optimal interpolation filter is linear and time invariant, even though this is not assumed a priori. We also show that the problem of estimating missing samples (optimal recovery), is equivalent to a problem of approximating the representer of a missing sample by a linear combination of the representers of the known samples. This approximation problem is shown to be the same as the minimization of an integrated squared error in the frequency domain, thus demonstrating the connection between optimal recovery, and a classical method of interpolation filter design.