Quantitative Fourier analysis of approximation techniques: Part I - Interpolators and projectors

We present a general Fourier-based method that pro, ides an accurate predic tion of the approximation error as a function-of the sampling step T, Our f ormalism applies to an extended class of convolution-based signal approxima tion techniques, which includes interpolation, generalized sampling with pr efiltering, and the projectors encountered in wavelet theory, We claim that are can predict the L-2-approximation error by integrating the spectrum of the function to approximate-not necessarily bandlimited-against a frequenc y kernel E(omega) that characterizes the approximation operator. This predi ction is easier yet more precise than was previously available, Our approac h has the remarkable property of providing a global error estimate that is the average of the true approximation error over all possible shifts of the input function. Our error prediction is exact for stationary processes, as well as for bandlimited signals. We apply this method to the comparison of standard interpolation and approximation techniques. Our method has interesting implications for approximation theory. In partic ular, we use our results to obtain some new asymptotic expansions of the er ror as T --> 0, as well as to derive improved upper bounds of the kind foun d in the Strang-Fix theory. We finally show how we can design quasi-interpo lators that are near optimal in the least-squares sense.