Quantitative Fourier Analysis of Approximation Techniques: Part I—Interpolators and Projectors

We present a general Fourier-based method that provides an accurate prediction of the approximation error as a function of the sampling step T . Our formalism applies to an extended class of convolution-based signal approximation techniques, which includes interpolation, generalized sampling with prefiltering, and the projectors encountered in wavelet theory. We claim that we can predict the L-approximation error by integrating the spectrum of the function to approximate—not necessarily bandlimited—against a frequency kernel E(!) that characterizes the approximation operator. This prediction is easier yet more precise than was previously available. Our approach 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 particular, we use our results to obtain some new asymptotic expansions of the error as T ! 0, as well as to derive improved upper bounds of the kind found in the Strang–Fix theory. We finally show how we can design quasi-interpolators that are near optimal in the least-squares sense.