Approximation Kernel Example

Comparing the sinc to the nearest-neighbor interpolation provides a good test case to grasp the predictive power of E. Since the Fourier transform of the sinc function is simply a rectangular function that takes a unit value in [— n, n] and is zero elsewhere, the denominator in (10) is unity for any frequency w (because a single term of the main domain contributes to the infinite sum). On the other hand, since the summation is performed over nonnull integers k e Z, there is no contributing term on the numerator side and E is zero in the main domain [— n, n]. By similar reasoning, it takes value 2 outside of the main domain. This corresponds to the well-known fact that a sinc synthesis function can represent a band-limited function with no error, and at the same time suffers a lot from aliasing when the function is not band-limited.

Nearest-neighbor interpolation is characterized by a rectangular synthesis function, the Fourier transform of which is a sinc function (this situation is the converse of the previous case). Unfortunately, expression (10) is now less manageable. We can nevertheless plot a numeric estimate of (9). The resulting approximation kernels are represented in Fig. 7.

At the origin, when w = 0, it is apparent from Fig. 7 that both sinc and nearest-neighbor interpolation produce no error; thus, they reproduce the constant. More generally, the degree

of "flatness" at the origin (the number of vanishing derivatives) directly gives the approximation order of the synthesis function — it being a straight line in the sine case, this flatness is infinite, and so is the approximation order. When w grows, interpolation by nearest-neighbor is less good than sinc interpolation, the latter being perfect up to Nyquist's frequency. Less known, but apparent from Fig. 7, is the fact that nearest-neighbor interpolation is indeed better than sinc for some (not all) of the part (if any) of the function to interpolate that is not band-limited.

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