## Reproduction of the Polynomials

We have already discussed the fact that the approximation order L is relevant only when the data are oversampled, and we mentioned that this is indeed broadly true for typical images. The new light brought by the Strang-Fix theory is that it is equivalent to think in terms of frequency contents or in terms of polynomials — apart from some technical details. Intuitively, it is reasonable to think that, when the data is smooth at scale h, we can model it by polynomials without introducing too much error. This has been formalized as the Weierstrass approximation theorem. What the Strang-Fix theory tells us is that there exists a precise relation between the approximation order and the maximal degree of the polynomials that the synthesis function can reproduce exactly. For example, we started this discussion on the theoretical assessment of the quality of any synthesis function by investigating the reproduction of the constant. We have now completed a full circle and are back to the reproduction of polynomials, of which the constant is but a particular case.

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FIGURE 6 Interpolant without regularity but with third-order approximation.

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FIGURE 6 Interpolant without regularity but with third-order approximation.

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