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  "body": "\n## Summary\n\nThe modulus of smoothness of order $k$ of a function measures the size of its $k$-th finite differences with small step.\nFor $k = 1$ in the uniform norm it is the [[modulus of continuity|AXCJBSQZWF]].\nHigher orders detect smoothness beyond Lipschitz, and they give the sharp form of the direct and inverse theorems of approximation by polynomials.\n\n## Explanations\n\n### L1 human\n\nThe first difference $f(x + h) - f(x)$ is small when $f$ changes little over a step $h$.\nThe second difference $f(x + 2h) - 2 f(x + h) + f(x)$ is small when $f$ is nearly a straight line over two steps; it vanishes for every linear function.\nThe modulus of smoothness records the largest such difference over all steps up to $t$.\nFor $f(x) = x$ the first-order modulus is $t$ and the second-order modulus is $0$.\n",
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